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Deck 9: Matrices and Determinants

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Question
Use Matrices and Gaussian Elimination to Solve Systems
x+y+z=2x−y+5z=−42x+y+z=7\begin{aligned}x + y + z & = 2 \\x - y + 5 z & = - 4 \\2 x + y + z & = 7\end{aligned}

A) {(5,−1,−2)}\{ ( 5 , - 1 , - 2 ) \}
B) {(5,−2,−1)}\{ ( 5 , - 2 , - 1 ) \}
C) {(−2,−1,5)}\{ ( - 2 , - 1,5 ) \}
D) {(−2,5,−1)}\{ ( - 2,5 , - 1 ) \}
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Question
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.
[11−11−301−44000161700017]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & - 3 \\ 0 & 1 & - 4 & 4 & 0 \\ 0 & 0 & 1 & 6 & 17 \\ 0 & 0 & 0 & 1 & 7 \end{array} \right]

A) {(93,−128,−25,7)}\{ ( 93 , - 128 , - 25,7 ) \}
В) {(−3,0,17,7)}\{ ( - 3,0,17,7 ) \}
C) {(−5,−1,10,6)}\{ ( - 5 , - 1,10,6 ) \}
D) {(7,−25,−128,93)}\{ ( 7 , - 25 , - 128,93 ) \}
Question
Perform Matrix Row Operations
[5−414−502−3−11−2−1]−5R1+R2\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }

A)
[5−414−3020−3−23−11−2−1]\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 30 & 20 & - 3 & - 23 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
B)
[5−41420−20717−11−2−1]\left[ \begin{array} { c r r | c } 5 & - 4 & 1 & 4 \\ 20 & - 20 & 7 & 17 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
C)
[30−4−919−502−3−11−2−1]\left[ \begin{array} { r r r | r } 30 & - 4 & - 9 & 19 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
D)
[−3020−3−23−502−3−11−2−1]\left[ \begin{array} { c r r | r } - 30 & 20 & - 3 & - 23 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
Question
Simplify Complex Rational Expressions
9x+5z=653y+6z=272x+8y+6z=42\begin{array} { r } 9 x + 5 z = 65 \\3 y + 6 z = 27 \\2 x + 8 y + 6 z = 42\end{array}

A)
[905650362728642]\left[ \begin{array} { l l l | l } 9 & 0 & 5 & 65 \\ 0 & 3 & 6 & 27 \\ 2 & 8 & 6 & 42 \end{array} \right]

B)

[902650382756642]\left[ \begin{array} { l l | l | l } 9 & 0 & 2 & 65 \\ 0 & 3 & 8 & 27 \\ 5 & 6 & 6 & 42 \end{array} \right]
C)
[950653602728642]\left[\begin{array}{lll|l}9 & 5 & 0 & 65 \\3 & 6 & 0 & 27 \\2 & 8 & 6 & 42\end{array}\right]
D)
[905036286]\left[\begin{array}{lll}9 & 0 & 5 \\0 & 3 & 6 \\2 & 8 & 6\end{array}\right]
Question
Simplify Complex Rational Expressions
x−4y+z=15y+8z=11z=14\begin{array} { r } x - 4 y + z = 15 \\y + 8 z = 11 \\z = 14\end{array}

A)
[1−41150181100114]\left[ \begin{array} { c c c | c } 1 & - 4 & 1 & 15 \\ 0 & 1 & 8 & 11 \\ 0 & 0 & 1 & 14 \end{array} \right]
B)

[0−40150081100014]\left[ \begin{array} { c c | c | c } 0 & - 4 & 0 & 15 \\ 0 & 0 & 8 & 11 \\ 0 & 0 & 0 & 14 \end{array} \right]
C)
[141150181100114]\quadD)[1−4115118111114]\left[ \begin{array} { l l l | l } 1 & 4 & 1 & 15 \\0 & 1 & 8 & 11 \\0 & 0 & 1 & 14\end{array} \right] \quadD) \left[ \begin{array} { r r r | r } 1 & - 4 & 1 & 15 \\1 & 1 & 8 & 11 \\1 & 1 & 14\end{array} \right]
Question
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.
[653−220744202]\left[ \begin{array} { r r r | r } 6 & 5 & 3 & - 2 \\ 2 & 0 & 7 & 4 \\ 4 & 2 & 0 & 2 \end{array} \right]

A) 6x+5y+3z=−26 x + 5 y + 3 z = - 2
2x+7z=42 x + 7 z = 4
4x+2y=24 x + 2 y = 2
B) 6x−5y+3z=−26 x - 5 y + 3 z = - 2
2x+7z=−42 x + 7 z = - 4
4x+2y=−24 x + 2 y = - 2
C) 6x+5y+3z=−26 x + 5 y + 3 z = - 2
2x+7z=42 x + 7 z = 4
4x+2z=24 x + 2 z = 2
D) 6x+5y+3z=−26 x + 5 y + 3 z = - 2
2x+y+7z=42 x + y + 7 z = 4
4x+2y+z=24 x + 2 y + z = 2
4x+2y+z=24 x + 2 y + z = 2
Question
Use Matrices and Gauss-Jordan Elimination to Solve Systems
x=−1−y−zx−y+4z=−63x+y=7−z\begin{array} { l } x = - 1 - y - z \\x - y + 4 z = - 6 \\3 x + y = 7 - z\end{array}

A) {(4,−2,−3)}\{ ( 4 , - 2 , - 3 ) \}
В) {(−2,−3,4)}\{ ( - 2 , - 3,4 ) \}
C) {(−3,−2,4)}\{ ( - 3 , - 2,4 ) \}
D) {(−3,4,−2)}\{ ( - 3,4 , - 2 ) \}
Question
Use Matrices and Gaussian Elimination to Solve Systems
x+y+z−w=62x−y+3z+4w=−44x+2y−z−w=−13−x−2y+4z+3w=12\begin{array} { l r } x + y + z - w = & 6 \\2 x - y + 3 z + 4 w = & - 4 \\4 x + 2 y - z - w = & - 13 \\- x - 2 y + 4 z + 3 w = & 12\end{array}

A) {(−4,3,5,−2)}\{ ( - 4,3,5 , - 2 ) \}
B) {(4,−3,−5,2)}\{ ( 4 , - 3 , - 5,2 ) \}
C) {(−14,13,15,−12)}\left\{ \left( - \frac { 1 } { 4 } , \frac { 1 } { 3 } , \frac { 1 } { 5 } , - \frac { 1 } { 2 } \right) \right\}
D) {(14,−13,−15,12)}\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 3 } , - \frac { 1 } { 5 } , \frac { 1 } { 2 } \right) \right\}
Question
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.
[15−8401−6−4001−7]\left[ \begin{array} { r r r | r } 1 & 5 & - 8 & 4 \\ 0 & 1 & - 6 & - 4 \\ 0 & 0 & 1 & - 7 \end{array} \right]

A) {(178,−46,−7)}\{ ( 178 , - 46 , - 7 ) \}
B) {(4,−4,−7)}\{ ( 4 , - 4 , - 7 ) \}
C) {(6,1,−8)}\{ ( 6,1 , - 8 ) \}
D) {(250,38,−7)}\{ ( 250,38 , - 7 ) \}
Question
Use Matrices and Gauss-Jordan Elimination to Solve Systems
−4x+7y−z=39x+6y+9z=62−5x+y+z=−1\begin{array} { r } - 4 x + 7 y - z = 39 \\x + 6 y + 9 z = 62 \\- 5 x + y + z = - 1\end{array}

A) {(2,7,2)}\{ ( 2,7,2 ) \}
B) {(2,2,7)}\{ ( 2,2,7 ) \}
C) {(−2,7,4)}\{ ( - 2,7,4 ) \}
D) {(4,7,−2)}\{ ( 4,7 , - 2 ) \}
Question
Perform Matrix Row Operations
[−20601045113−302−7421]15R1\left[ \begin{array} { r r r | r } - 20 & 60 & 10 & 45 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right] \frac { 1 } { 5 } \mathrm { R } _ { 1 }

A)
[−41229113−302−7421]\left[ \begin{array} { c c r | r } - 4 & 12 & 2 & 9 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right]
B)

[−2060104515135−3502−7421]\left[ \begin{array} { c c r | r } - 20 & 60 & 10 & 45 \\\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \\2 & - 7 & 4 & 21\end{array} \right]
C)
[−412245113−302−7421]\quadD)[−4122915135−35025−7545215]\left[ \begin{array} { r r r | r } - 4 & 12 & 2 & 45 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right] \quadD) \left[ \begin{array} { r r r | r } - 4 & 12 & 2 & 9 \\\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \\\frac { 2 } { 5 } - \frac { 7 } { 5 } & \frac { 4 } { 5 } & \frac { 21 } { 5 }\end{array} \right]
Question
Use Matrices and Gauss-Jordan Elimination to Solve Systems
−3x−y−7z=−763x+3y−6z=−27−9x−2y+z=−23\begin{aligned}- 3 x - y - 7 z & = - 76 \\3 x + 3 y - 6 z & = - 27 \\- 9 x - 2 y + z & = - 23\end{aligned}

A) {(2,7,9)}\{ ( 2,7,9 ) \}
B) {(2,9,7)}\{ ( 2,9,7 ) \}
C) {(−2,7,4)}\{ ( - 2,7,4 ) \}
D) {(4,7,−2)}\{ ( 4,7 , - 2 ) \}
Question
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.
[13211201−12−80014]\left[ \begin{array} { r r r | r } 1 & \frac { 3 } { 2 } & 1 & \frac { 1 } { 2 } \\ 0 & 1 & - \frac { 1 } { 2 } & - 8 \\ 0 & 0 & 1 & 4 \end{array} \right]

A) {(112,−6,4)}\left\{ \left( \frac { 11 } { 2 } , - 6,4 \right) \right\}
B) {(12,−8,4)}\left\{ \left( \frac { 1 } { 2 } , - 8,4 \right) \right\}
C) {(−3,−172,3)}\left\{ \left( - 3 , - \frac { 17 } { 2 } , 3 \right) \right\}
D) {(−212,−6,4)}\left\{ \left( - \frac { 21 } { 2 } , - 6,4 \right) \right\}
Question
Simplify Complex Rational Expressions
12x+7y−8z+w=15y+z=−6x−y−2z=−53x−3y+4z=5\begin{array} { r } 12 x + 7 y - 8 z + w = 1 \\5 y + z = - 6 \\x - y - 2 z = - 5 \\3 x - 3 y + 4 z = 5\end{array}

A)
[127−8110510−61−1−20−53−3405]\left[ \begin{array} { l r r r | r } 12 & 7 & - 8 & 1 & 1 \\0 & 5 & 1 & 0 & - 6 \\1 & - 1 & - 2 & 0 & - 5 \\3 & - 3 & 4 & 0 & 5\end{array} \right]
B)
[127−8105161−1−2−53−345]\left[ \begin{array} { l r r | r } 12 & 7 & - 8 & 1 \\0 & 5 & 1 & 6 \\1 & - 1 & - 2 & - 5 \\3 & - 3 & 4 & 5\end{array} \right]
C)
[1201975−1−3−81−2410001−6−55]\left[ \begin{array} { l r r | r } 12 & 0 & 1 & 9 \\7 & 5 & - 1 & - 3 \\- 8 & 1 & - 2 & 4 \\1 & 0 & 0 & 0 \\1 & - 6 & - 5 & 5\end{array} \right]
D)
[1278110510−61120−533405]\left[ \begin{array} { l l l l | r } 12 & 7 & 8 & 1 & 1 \\0 & 5 & 1 & 0 & - 6 \\1 & 1 & 2 & 0 & - 5 \\3 & 3 & 4 & 0 & 5\end{array} \right]
Question
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.
[510212−1810−69006−11030−3−4]\left[ \begin{array} { r r r r | r } 5 & 1 & 0 & 2 & 12 \\- 1 & 8 & 1 & 0 & - 6 \\9 & 0 & 0 & 6 & - 11 \\0 & 3 & 0 & - 3 & - 4\end{array} \right]

A)
5x+y+2w=125 x + y + 2 w = 12
−x+8y+z=−6- x + 8 y + z = - 6
9x+6w=−119 x + 6 w = - 11
3y−3w=−43 y - 3 w = - 4
B)
5x+y+z+2w=125 x + y + z + 2 w = 12
−x+8y+z+w=−6- x + 8 y + z + w = - 6
9x+y+z+6w=−119 x + y + z + 6 w = - 11
x+3y+z−3w=−4x + 3 y + z - 3 w = - 4
C)
5x+y+2w=125 x + y + 2 w = 12
x+8y+z=−6x + 8 y + z = - 6
9x+6w=−119 x + 6 w = - 11
3y+3w=−43 y + 3 w = - 4
D)
5x+y+2z=125 x + y + 2 z = 12
−x+8y+z=−6- x + 8 y + z = - 6
9x+6y=−119 x + 6 y = - 11
3x−3y=−43 x - 3 y = - 4
Question
Use Matrices and Gaussian Elimination to Solve Systems
x−y+4z=152x+z=3x+2y+z=−3\begin{aligned}x - y + 4 z & = 15 \\2 x + z & = 3 \\x + 2 y + z & = - 3\end{aligned}

A) {(0,−3,3)}\{ ( 0 , - 3,3 ) \}
B) {(0,3,−3)}\{ ( 0,3 , - 3 ) \}
C) {(3,−3,0)}\{ ( 3 , - 3,0 ) \}
D) {(3,0,−3)}\{ ( 3,0 , - 3 ) \}
Question
Perform Matrix Row Operations
[11−1130−43−5030−1−41−3301−4]−2R1+R33R1+R4\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 3 & 0 & - 1 & - 4 & 1 \\ - 3 & 3 & 0 & 1 & - 4 \end{array} \right] \quad \begin{array} { r } - 2 R _ { 1 } + R _ { 3 } \\ 3 R _ { 1 } + R _ { 4 } \end{array}

A)
[11−1130−43−501−21−6−506−345]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 1 & - 2 & 1 & - 6 & - 5 \\ 0 & 6 & - 3 & 4 & 5 \end{array} \right]
B)
[11−1130−43−501−21−6−5−3301−4]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 1 & - 2 & 1 & - 6 & - 5 \\ - 3 & 3 & 0 & 1 & - 4 \end{array} \right]
C)
[11−1130−43−5052−3−2706−345]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 5 & 2 & - 3 & - 2 & 7 \\ 0 & 6 & - 3 & 4 & 5 \end{array} \right]
D)
[11−1130−43−501−21−6−106−344]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 1 & - 2 & 1 & - 6 & - 1 \\ 0 & 6 & - 3 & 4 & 4 \end{array} \right]
Question
Use Matrices and Gaussian Elimination to Solve Systems
5x−y−3z=−83x−8z=−217y+z=31\begin{aligned}5 x - y - 3 z & = - 8 \\3 x - 8 z & = - 21 \\7 y + z & = 31\end{aligned}

A) {(1,4,3)}\{ ( 1,4,3 ) \}
B) {(1,3,4)}\{ ( 1,3,4 ) \}
C) {(−1,4,2)}\{ ( - 1,4,2 ) \}
D) {(−1,2,4)}\{ ( - 1,2,4 ) \}
Question
Simplify Complex Rational Expressions
2x+6y+9z=667x+6y+8z=779x+8y−2z=51\begin{array} { l } 2 x + 6 y + 9 z = 66 \\7 x + 6 y + 8 z = 77 \\9 x + 8 y - 2 z = 51\end{array}

A)
[269667687798−251]\left[ \begin{array} { r r r | r } 2 & 6 & 9 & 66 \\ 7 & 6 & 8 & 77 \\ 9 & 8 & - 2 & 51 \end{array} \right]

B)

[279666687798−251]\left[ \begin{array} { r r r | r } 2 & 7 & 9 & 66 \\ 6 & 6 & 8 & 77 \\ 9 & 8 & - 2 & 51 \end{array} \right]
C)
[669627786751−289]\left[\begin{array}{rrr|r}66 & 9 & 6 & 2 \\77 & 8 & 6 & 7 \\51 & -2 & 8 & 9\end{array}\right]

D)
[26976898−2]\left[\begin{array}{rrr}2 & 6 & 9 \\7 & 6 & 8 \\9 & 8 & -2\end{array}\right]
Question
Use Matrices and Gaussian Elimination to Solve Systems
3x+5y−2w=−132x+7z−w=−14y+3z+3w=1−x+2y+4z=−5\begin{aligned}3 x + 5 y - 2 w = & - 13 \\2 x + 7 z - w = & - 1 \\4 y + 3 z + 3 w = & 1 \\- x + 2 y + 4 z = & - 5\end{aligned}

A) {(1,−2,0,3)}\{ ( 1 , - 2,0,3 ) \}
B) {(43,−1320,0,52)}\left\{ \left( \frac { 4 } { 3 } , - \frac { 13 } { 20 } , 0 , \frac { 5 } { 2 } \right) \right\}
C) {(34,−2,0,34)}\left\{ \left( \frac { 3 } { 4 } , - 2,0 , \frac { 3 } { 4 } \right) \right\}
D) {(−1,−2013,0,25)}\left\{ \left( - 1 , - \frac { 20 } { 13 } , 0 , \frac { 2 } { 5 } \right) \right\}
Question
Use Matrices and Gauss-Jordan Elimination to Solve Systems
x+y−z+w=−53x−y+3z−2w=7−2x+2y+z−w=16−x−2y−3z+3w=−22\begin{array} { c } x + y - z + w = - 5 \\3 x - y + 3 z - 2 w = 7 \\- 2 x + 2 y + z - w = 16 \\- x - 2 y - 3 z + 3 w = - 22\end{array}

A) {(−2,3,4,−2)}\{ ( - 2,3,4 , - 2 ) \}
B) {(−2,−3,5,12)}\left\{ \left( - 2 , - 3,5 , \frac { 1 } { 2 } \right) \right\}
C) {(2,−3,−4,−2)}\{ ( 2 , - 3 , - 4 , - 2 ) \}
D) {(12,−13,−14,−12)}\left\{ \left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } , - \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right) \right\}
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z+w=73x−2z+5w=11−4x+3y+w=4−x−y−z−w=6\begin{aligned}x + y + z + w & = 7 \\3 x - 2 z + 5 w & = 11 \\- 4 x + 3 y + w & = 4 \\- x - y - z - w & = 6\end{aligned}

A) ∅\varnothing
B) {(32,1,13,−2)}\left\{ \left( \frac { 3 } { 2 } , 1 , \frac { 1 } { 3 } , - 2 \right) \right\}
C) {(74,−12,5,−16)}\left\{ \left( \frac { 7 } { 4 } , - \frac { 1 } { 2 } , 5 , - \frac { 1 } { 6 } \right) \right\}
D) {(−11,719,619,−4)}\left\{ \left( - 11 , \frac { 7 } { 19 } , \frac { 6 } { 19 } , - 4 \right) \right\}
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
4x−y+3z=12x+4y+6z=−325x+3y+9z=20\begin{aligned}4 x - y + 3 z & = 12 \\x + 4 y + 6 z & = - 32 \\5 x + 3 y + 9 z & = 20\end{aligned}

A) ∅\varnothing
В) {(2,−7,−1)}\{ ( 2 , - 7 , - 1 ) \}
C) {(8,−7,−2)}\{ ( 8 , - 7 , - 2 ) \}
D) {(−8,−7,9)}\{ ( - 8 , - 7,9 ) \}
Question
Apply Gaussian Elimination to Systems with More Variables than Equations
x+y+z=92x−3y+4z=7\begin{array} { l } x + y + z = 9 \\2 x - 3 y + 4 z = 7\end{array}

A) {(−75z+345,25z+115,z)}\left\{ \left( - \frac { 7 } { 5 } \mathrm { z } + \frac { 34 } { 5 } , \frac { 2 } { 5 } \mathrm { z } + \frac { 11 } { 5 } , \mathrm { z } \right) \right\}
B) {(35z+165,−85z+295,z)}\left\{ \left( \frac { 3 } { 5 } z + \frac { 16 } { 5 } , - \frac { 8 } { 5 } z + \frac { 29 } { 5 } , z \right) \right\}
C) {(275,135,1)}\left\{ \left( \frac { 27 } { 5 } , \frac { 13 } { 5 } , 1 \right) \right\}
D) ∅\varnothing
Question
Write a system of linear equations in three variables, and then use matrices to solve the system.
Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake to 103 g protein, 93 g fat, and 135 g carbohydrate. According to the health conscious hostess, the marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g carbohydrate. How many of each snack can he eat to obtain his goal?

A) 7 mushrooms; 4 meatballs; 2 eggs
B) 4 mushrooms; 2 meatballs; 7 eggs
C) 2 mushrooms; 7 meatballs; 4 eggs
D) 8 mushrooms; 5 meatballs; 3 eggs
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
5x+2y+z=−112x−3y−z=177x−y=12\begin{aligned}5 x + 2 y + z & = - 11 \\2 x - 3 y - z & = 17 \\7 x - y & = 12\end{aligned}

A) ∅\varnothing
B) {(0,−6,1)}\{ ( 0 , - 6,1 ) \}
C) {(−2,0,−1)}\{ ( - 2,0 , - 1 ) \}
D) {(1,−5,0)}\{ ( 1 , - 5,0 ) \}
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
3x−2y+2z−w=24x+y+z+6w=8−3x+2y−2z+w=55x+3z−2w=1\begin{array} { r } 3 x - 2 y + 2 z - w = 2 \\4 x + y + z + 6 w = 8 \\- 3 x + 2 y - 2 z + w = 5 \\5 x + 3 z - 2 w = 1\end{array}

A) ∅\varnothing
B) {(2,0,−337,937)}\left\{ \left( 2,0 , - \frac { 3 } { 37 } , \frac { 9 } { 37 } \right) \right\}
C) {(12,0,−373,379)}\left\{ \left( \frac { 1 } { 2 } , 0 , - \frac { 37 } { 3 } , \frac { 37 } { 9 } \right) \right\}
D) {(1,−13,49,6)}\left\{ \left( 1 , - \frac { 1 } { 3 } , \frac { 4 } { 9 } , 6 \right) \right\}
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+3y+2z=114y+9z=−12x+7y+11z=−1\begin{array} { r } x + 3 y + 2 z = 11 \\4 y + 9 z = - 12 \\x + 7 y + 11 z = - 1\end{array}

A) {(19z4+20,−9z4−3,z)}\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } - 3 , z \right) \right\}
B) {(19z4+20,−9z4+3,z)}\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}
C) {(19z4+20,9z4+3,z)}\left\{ \left( \frac { 19 z } { 4 } + 20 , \frac { 9 z } { 4 } + 3 , z \right) \right\}
D) {(−19z4+20,−9z4+3,z)}\left\{ \left( - \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+8y+8z=87x+7y+z=18x+15y+9z=−9\begin{aligned}x + 8 y + 8 z & = 8 \\7 x + 7 y + z & = 1 \\8 x + 15 y + 9 z & = - 9\end{aligned}

A) ∅\varnothing
B) {(0,0,1)}\{ ( 0,0,1 ) \}
C) {(1,−1,1)}\{ ( 1 , - 1,1 ) \}
D) {(−1,0,1)}\{ ( - 1,0,1 ) \}
Question
Write a system of linear equations in three variables, and then use matrices to solve the system.
A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A tree takes 14 hours to prepare, 3 hours to paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 15 hours to paint, and 7 hours to fire. If the workshop has 85 hours for prep time, 56 hours for painting, and 100 hours for firing, how many of each can be made?

A) 7 wreaths; 4 trees; 2 sleighs
B) 4 wreaths; 2 trees; 7 sleighs
C) 2 wreaths; 7 trees; 4 sleighs
D) 8 wreaths; 5 trees; 3 sleighs
Question
Write a system of linear equations in three variables, and then use matrices to solve the system.
There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sales by age group for marketing purposes. The percentage of 36- to 59-year-old buyers and the percentage of buyers 60 and older combined exceeds the percentage of buyers 35 and younger by 40%. If the percentage of buyers in the oldest group is doubled, it is 34% less than the percentage of users in the middle group. Find the percentage of buyers in each of the three age groups.

A) 30% 35 and younger; 58% 36-59 year olds; 12% 60 and older
B) 32% 35 and younger; 55% 36-59 year olds; 13% 60 and older
C) 24% 35 and younger; 60% 36-59 year olds; 16% 60 and older
D) 12% 35 and younger; 58% 36-59 year olds; 30% 60 and older
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z=92x−3y+4z=7x−4y+3z=−2\begin{array} { r r } x + y + z & = 9 \\2 x - 3 y + 4 z & = 7 \\x - 4 y + 3 z & = - 2\end{array}

A) {(−7z5+345,2z5+115,z)}\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}
B) {(z5+345,2z5+115,z)}\left\{ \left( \frac { z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}
C) {(−7z5+345,2z5−115,z)}\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}
D) {(7z5+345,2z5−115,z)}\left\{ \left( \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , \mathrm { z } \right) \right\}
Question
Apply Gaussian Elimination to Systems with More Variables than Equations
x+y+z=7x−y+2z=7\begin{array} { l } x + y + z = 7 \\x - y + 2 z = 7\end{array}

A) {(−32z+7,12z,z)}\left\{ \left( - \frac { 3 } { 2 } z + 7 , \frac { 1 } { 2 } z , z \right) \right\}
B) {(−3z+14,2z−7,z)}\{ ( - 3 z + 14,2 z - 7 , z ) \}
C) {(4,1,2)}\{ ( 4,1,2 ) \}
D) {(8,−3,2)}\{ ( 8 , - 3,2 ) \}
Question
Apply Gaussian Elimination to Systems with More Variables than Equations
3x+y+z−2w=102x+3y+3z+w=−52x+y+4z+11w=11\begin{array} { r } 3 x + y + z - 2 w = 10 \\2 x + 3 y + 3 z + w = - 5 \\2 x + y + 4 z + 11 w = 11\end{array}

A) {(w+5,3w−7,−4w+2,w)}\{ ( w + 5,3 w - 7 , - 4 w + 2 , w ) \}
B) {(2w+3,6w−7,−10w+8,w)}\{ ( 2 w + 3,6 w - 7 , - 10 w + 8 , w ) \}
C) {(6,−4,−2,1)}\{ ( 6 , - 4 , - 2,1 ) \}
D) {(7,−1,−6,2)}\{ ( 7 , - 1 , - 6,2 ) \}
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z+w=83x+2y+z+4w=214x+4y+5z+8w=302x+3y+6z+9w=15\begin{array} { r } x + y + z + w = 8 \\3 x + 2 y + z + 4 w = 21 \\4 x + 4 y + 5 z + 8 w = 30 \\2 x + 3 y + 6 z + 9 w = 15\end{array}

A) {(−6w+3,9w+7,−4w−2,w)}\{ ( - 6 w + 3,9 w + 7 , - 4 w - 2 , w ) \}
B) {(5w+11,−3w−7,−3w+4,w)}\{ ( 5 w + 11 , - 3 w - 7 , - 3 w + 4 , w ) \}
C) {(−3,16,−6,1)}\{ ( - 3,16 , - 6,1 ) \}
D) ∅\varnothing
Question
Apply Gaussian Elimination to Systems with More Variables than Equations
5x−y+z=87x+y+z=6\begin{array} { l } 5 x - y + z = 8 \\7 x + y + z = 6\end{array}

A) {(−16z+76,16z−136,z)}\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}
B) {(−z+3,4z+7,z)}\{ ( - z + 3,4 z + 7 , z ) \}
C) {(16z+76,16z,z)}\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}
D) ∅\varnothing
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x−y+z−w=10−2x+3y+5w=−28x+2y+8z+3w=−10x−4y−6z−5w=30\begin{array} { r } x - y + z - w = 10 \\- 2 x + 3 y + 5 w = - 28 \\x + 2 y + 8 z + 3 w = - 10 \\x - 4 y - 6 z - 5 w = 30\end{array}

A) {(−17w−10,−13w−16,5w+4,w)}\{ ( - 17 w - 10 , - 13 w - 16,5 w + 4 , w ) \}
B) {(3w−2,−8w+3,4w+9,w)}\{ ( 3 w - 2 , - 8 w + 3,4 w + 9 , w ) \}
C) {(24,10,−6,−2)}\{ ( 24,10 , - 6 , - 2 ) \}
D) ∅\varnothing
Question
Use Matrices and Gauss-Jordan Elimination to Solve Systems
3x+5y+2w=−122x+6z−w=−5−2y+3z−3w=−3−x+2y+4z+w=−2\begin{aligned}3 x + 5 y + 2 w & = - 12 \\2 x + 6 z - w & = - 5 \\- 2 y + 3 z - 3 w & = - 3 \\- x + 2 y + 4 z + w & = - 2\end{aligned}

A) {(−1,−3,0,3)}\{ ( - 1 , - 3,0,3 ) \}
B) {(1,−3,0,3)}\{ ( 1 , - 3,0,3 ) \}
C) {(−1,3,0,−3)}\{ ( - 1,3,0 , - 3 ) \}
D) {(1,3,0,−3)}\{ ( 1,3,0 , - 3 ) \} -
Question
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z=7x−y+2z=72x+3z=14\begin{array} { r } x + y + z = 7 \\x - y + 2 z = 7 \\2 x + 3 z = 14\end{array}

A) {(−3z2+7,z2,z)}\left\{ \left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right) \right\}
B) {(−3z2−7,z2,z)}\left\{ \left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right) \right\}
C) {(−3z2+7,2z,z)}\left\{ \left( - \frac { 3 z } { 2 } + 7,2 z , z \right) \right\}
D) {(−3z2−7,2z,z)}\left\{ \left( - \frac { 3 z } { 2 } - 7,2 z , z \right) \right\}
Question
Write a system of linear equations in three variables, and then use matrices to solve the system.
The table below shows the number of birds for three selected years after an endangered species protection program was started. x (Number of years after 1980) 1510y (Number of birds) 43139349\begin{array} { l | c c c } \mathrm { x } \text { (Number of years after 1980) } & 1 & 5 & 10 \\\hline \mathrm { y } \text { (Number of birds) } & 43 & 139 & 349\end{array} Use the quadratic function y=ax2+bx+cy = a x ^ { 2 } + b x + c to model the data. Solve the system of linear equations involving a,ba , b , and cc using matrices. Find the equation that models the data.

A) y=2x2+12x+29y = 2 x ^ { 2 } + 12 x + 29
B) y=3x2+24x+24y = 3 x ^ { 2 } + 24 x + 24
C) y=4x2−12x+32y = 4 x ^ { 2 } - 12 x + 32
D) y=4x2−36x+25y = 4 x ^ { 2 } - 36 x + 25
Question
Perform Scalar Multiplication
Let A=[−3602]\mathrm { A } = \left[ \begin{array} { r r } - 3 & 6 \\ 0 & 2 \end{array} \right] . Find 4A.

A)
[−122408]\left[ \begin{array} { r r } - 12 & 24 \\0 & 8\end{array} \right]
B)
[−122402]\left[\begin{array}{rr}-12 & 24 \\0 & 2\end{array}\right]
C)
[−12602]\left[\begin{array}{rr}-12 & 6 \\0 & 2\end{array}\right]
D)
[11046]\left[\begin{array}{rr}1 & 10 \\4 & 6\end{array}\right]
Question
Add and Subtract Matrices
Let A=[−1031]\mathrm { A } = \left[ \begin{array} { r r } - 1 & 0 \\ 3 & 1 \end{array} \right] and B=[−1331]\mathrm { B } = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 1 \end{array} \right] . Find A−B\mathrm { A } - \mathrm { B } .

A)
[0−300]\left[ \begin{array} { r r } 0 & - 3 \\ 0 & 0 \end{array} \right]
B)
[−2362]\left[ \begin{array} { r } - 23 \\ 62 \end{array} \right]
C)
[0300]\left[ \begin{array} { l l } 0 & 3 \\ 0 & 0 \end{array} \right]
D)
[−3][ - 3 ]
Question
Understand What is Meant by Equal Matrices
[x1]=[−1y]\left[ \begin{array} { c } x \\1\end{array} \right] = \left[ \begin{array} { l } - 1 \\y\end{array} \right]

A) x=−1;y=1x = - 1 ; y = 1
B) x=1;y=−1x = 1 ; y = - 1
C) x=−1;y=−1x = - 1 ; y = - 1
D) x=1;y=1x = 1 ; y = 1
Question
Understand What is Meant by Equal Matrices
[x+3y+475]=[7−37z]\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 5\end{array} \right] = \left[ \begin{array} { r r } 7 & - 3 \\7 & z\end{array} \right]
B) x=−4;y=7;z=−5x = - 4 ; y = 7 ; z = - 5

A) x=4;y=−7;z=5x = 4 ; y = - 7 ; z = 5
D) x=4;y=5;z=7x = 4 ; y = 5 ; z = 7
C) x=7;y=−3;z=5x = 7 ; y = - 3 ; z = 5
Question
Understand What is Meant by Equal Matrices
[xy+69z3]=[88903]\left[ \begin{array} { r r } x & y + 6 \\9 z & 3\end{array} \right] = \left[ \begin{array} { c c } 8 & 8 \\90 & 3\end{array} \right]

A) x=8;y=2;z=10x = 8 ; y = 2 ; z = 10
B) x=8;y=8;z=90x = 8 ; y = 8 ; z = 90
C) x=8;y=3;z=8x = 8 ; y = 3 ; z = 8
D) x=3;y=14;z=810x = 3 ; y = 14 ; z = 810
Question
Matrix Operations and Their Applications
1 Use Matrix Notation
[3767−33−e−15−13π−69−121281396−15];a34\left[ \begin{array} { c c c c c } 3 & 7 & 6 & 7 & - 3 \\3 & - e & - 15 & - 13 & \pi \\- 6 & 9 & - 12 & 12 & 8 \\\frac { 1 } { 3 } & 9 & 6 & - 1 & 5\end{array} \right] ; a _ { 34 }

A) 4×5;124 \times 5 ; 12
B) 5×4;65 \times 4 ; 6
C) 20;820 ; 8
D) 4×4;−124 \times 4 ; - 12
Question
Apply Gaussian Elimination to Systems with More Variables than Equations
2x+y+2z−4w=10x+3y+2z−11w=173x+y+7z−21w=0\begin{aligned}2 x + y + 2 z - 4 w & = 10 \\x + 3 y + 2 z - 11 w & = 17 \\3 x + y + 7 z - 21 w & = 0\end{aligned}

A) {(−3w+5,2w+6,4w−3,w)}\{ ( - 3 w + 5,2 w + 6,4 w - 3 , w ) \}
B) {(3w+5,6w+6,−4w−3,w)}\{ ( 3 w + 5,6 w + 6 , - 4 w - 3 , w ) \}
C) {(w+5,8w+4,−3w−2,w)}\{ ( w + 5,8 w + 4 , - 3 w - 2 , w ) \}
D) {(w−5,8w−4,−3w+2,w)}\{ ( w - 5,8 w - 4 , - 3 w + 2 , w ) \}
Question
Perform Scalar Multiplication
Let B=[−166−3]B = \left[ \begin{array} { l l l l } - 1 & 6 & 6 & - 3 \end{array} \right] . Find −3B- 3 B .

A) [3−18−18\left[ \begin{array} { l l l } 3 & - 18 & - 18 \end{array} \right. ]
B) [366\left[ \begin{array} { l l l } 3 & 6 & 6 \end{array} \right. ]
C) [−31818−9][ - 3 18 18 - 9 ]
D) [−344−5][ - 3 44 - 5 ]
Question
Perform Scalar Multiplication
Let A=[3324]\mathrm { A } = \left[ \begin{array} { l l } 3 & 3 \\ 2 & 4 \end{array} \right] and B=[04−16]\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right] . Find 3 A+B3 \mathrm {~A} + \mathrm { B } .

A)
[913518]\left[ \begin{array} { l l } 9 & 13 \\ 5 & 18 \end{array} \right]
B)
[921330]\left[ \begin{array} { l l } 9 & 21 \\ 3 & 30 \end{array} \right]
C)
[913110]\left[ \begin{array} { l l } 9 & 13 \\ 1 & 10 \end{array} \right]
D)
[97510]\left[ \begin{array} { c c } 9 & 7 \\ 5 & 10 \end{array} \right]
Question
Add and Subtract Matrices
Let A=[24142−1−123]\mathrm { A } = \left[ \begin{array} { r r r } 2 & 4 & 1 \\ 4 & 2 & - 1 \\ - 1 & 2 & 3 \end{array} \right] and B=[3−41−10203−2]\mathrm { B } = \left[ \begin{array} { r r r } 3 & - 4 & 1 \\ - 1 & 0 & 2 \\ 0 & 3 & - 2 \end{array} \right] . Find A+B\mathrm { A } + \mathrm { B } .

A)
[502321−151]\left[ \begin{array} { r r r } 5 & 0 & 2 \\ 3 & 2 & 1 \\ - 1 & 5 & 1 \end{array} \right]
B)
[3−82521−151]\left[ \begin{array} { r r r } 3 & - 8 & 2 \\ 5 & 2 & 1 \\ - 1 & 5 & 1 \end{array} \right]
C)
[3−8232−3151]\left[ \begin{array} { r r r } 3 & - 8 & 2 \\ 3 & 2 & - 3 \\ 1 & 5 & 1 \end{array} \right]
D)
[50252−3151]\left[ \begin{array} { r r r } 5 & 0 & 2 \\ 5 & 2 & - 3 \\ 1 & 5 & 1 \end{array} \right]
Question
Add and Subtract Matrices
Let A=[−5125]\mathrm { A } = \left[ \begin{array} { r r } - 5 & 1 \\ 2 & 5 \end{array} \right] and B=[624−2]\mathrm { B } = \left[ \begin{array} { r r } 6 & 2 \\ 4 & - 2 \end{array} \right] . Find A+B\mathrm { A } + \mathrm { B } .

A)
[1363]\left[\begin{array}{ll}1 & 3 \\6 & 3\end{array}\right]
B)
[34−13]\left[\begin{array}{r}34 \\-13\end{array}\right]
C)
[1−3−1−7]\left[\begin{array}{rr}1 & -3 \\-1 & -7\end{array}\right]

D)
[13][ 13 ]
Question
Add and Subtract Matrices
Let A=[4−1028−12−3459]\mathrm { A } = \left[ \begin{array} { r r r } 4 & - 10 & 2 \\ 8 & - 12 & - 3 \\ 4 & 5 & 9 \end{array} \right] and B=[010340−5−58−5]\mathrm { B } = \left[ \begin{array} { r r r } 0 & 10 & 3 \\ 4 & 0 & - 5 \\ - 5 & 8 & - 5 \end{array} \right] . Find A−B\mathrm { A } - \mathrm { B }

A)
[4−20−14−1229−314]\left[ \begin{array} { r r r } 4 & - 20 & - 1 \\ 4 & - 12 & 2 \\ 9 & - 3 & 14 \end{array} \right]
B)
[40−14−1221−34]\left[ \begin{array} { r r r } 4 & 0 & - 1 \\ 4 & - 12 & 2 \\ 1 & - 3 & 4 \end{array} \right]
C)
[420−14−1229314]\left[ \begin{array} { r r r } 4 & 20 & - 1 \\ 4 & - 12 & 2 \\ 9 & 3 & 14 \end{array} \right]
D)
[40512−12−8−1134]\left[ \begin{array} { r r r } 4 & 0 & 5 \\ 12 & - 12 & - 8 \\ - 1 & 13 & 4 \end{array} \right]
Question
Add and Subtract Matrices
A)
[−81−1705−6]\left[ \begin{array} { r r } - 8 & 1 \\- 17 & 0 \\5 & - 6\end{array} \right]

A)
[−81−1785−6]\left[ \begin{array} { r r } -8 & 1 \\ -17 & 8\\ 5 & -6 \end{array} \right]

B)
[1478111]\left[ \begin{array} { r r } 1 & 4 \\ 7 & 8 \\ 11 & 1 \end{array} \right]
C)
[11705−2]\left[ \begin{array} { r r } 1 & 1 \\ 7 & 0 \\ 5 & - 2 \end{array} \right]
D)
[3−270−56]\left[ \begin{array} { r r } 3 & - 2 \\ 7 & 0 \\ - 5 & 6 \end{array} \right]
Question
Solve Problems Involving Systems Without Unique Solutions
The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If construction limits z to t cars per minute, how many cars per minute must pass through the other intersections to keep traffic moving? <strong>Solve Problems Involving Systems Without Unique Solutions The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If construction limits z to t cars per minute, how many cars per minute must pass through the other intersections to keep traffic moving? </strong> A) t + 8 cars/min between I _ { 2 } and I _ { 1 } ; t + 3 cars/min between I _ { 1 } and I _ { 3 } B) t + 1 cars/min between I _ { 2 } and I _ { 1 } ; t + 4 cars/min between I _ { 1 } and I _ { 3 } C) t - 2 cars/min between I _ { 2 } and I _ { 1 } ; t + 1 cars/min between I _ { 1 } and I _ { 3 } D) t + 2 cars / \mathrm { min } between \mathrm { I } _ { 2 } and \mathrm { I } _ { 1 } ; \mathrm { t } - 3 cars/min between \mathrm { I } _ { 1 } and \mathrm { I } _ { 3 } <div style=padding-top: 35px>

A) t+8t + 8 cars/min between I2I _ { 2 } and I1;t+3I _ { 1 } ; t + 3 cars/min between I1I _ { 1 } and I3I _ { 3 }
B) t+1t + 1 cars/min between I2I _ { 2 } and I1;t+4I _ { 1 } ; t + 4 cars/min between I1I _ { 1 } and I3I _ { 3 }
C) t−2t - 2 cars/min between I2I _ { 2 } and I1;t+1I _ { 1 } ; t + 1 cars/min between I1I _ { 1 } and I3I _ { 3 }
D) t+2t + 2 cars /min/ \mathrm { min } between I2\mathrm { I } _ { 2 } and I1;t−3\mathrm { I } _ { 1 } ; \mathrm { t } - 3 cars/min between I1\mathrm { I } _ { 1 } and I3\mathrm { I } _ { 3 }
Question
Matrix Operations and Their Applications
1 Use Matrix Notation
[−82−105−9−24];a12\left[ \begin{array} { r c c c } - 8 & 2 & - 1 & 0 \\5 & - 9 & - 2 & 4\end{array} \right] ; \mathrm { a } _ { 12 }

A) 2×4;22 \times 4 ; 2
B) 4×2;24 \times 2 ; 2
C) 2×4;52 \times 4 ; 5
D) 4×2;54 \times 2 ; 5
Question
Understand What is Meant by Equal Matrices
[6−8−79]=[xy−7z]\left[ \begin{array} { r r } 6 & - 8 \\- 7 & 9\end{array} \right] = \left[ \begin{array} { r r } x y \\- 7 & z\end{array} \right]

A) x=6;y=−8;z=9x = 6 ; y = - 8 ; z = 9
B) x=6;y=−8;z=−7x = 6 ; y = - 8 ; z = - 7
C) x=6;y=−7;z=9x = 6 ; y = - 7 ; z = 9
D) x=−8;y=6;z=9x = - 8 ; y = 6 ; z = 9
Question
Solve Problems Involving Systems Without Unique Solutions
The nutritional content per ounce for three foods is given in the table below.  Fat (g/oz) Protein (g/oz) Fiber (g/oz) Food A 241 Food B 121 Food C 8165\begin{array}{l|ccc} & \text { Fat }(\mathrm{g} / \mathrm{oz}) & \text { Protein }(\mathrm{g} / \mathrm{oz}) & \text { Fiber }(\mathrm{g} / \mathrm{oz}) \\\hline \text { Food A } & 2 & 4 & 1 \\\text { Food B } & 1 & 2 & 1 \\\text { Food C } & 8 & 16 & 5\end{array}

What combination of these foods can provide exactly 14 grams of fat, 27 grams of protein, and 10 grams of fiber?

A) No possible combination of these foods
B) 3oz3 \mathrm { oz } of Food A; 5oz5 \mathrm { oz } of Food B; 1oz1 \mathrm { oz } of Food CC
C) 7 oz of Food A; 7 oz of Food B; 1 oz of Food CC
D) 4oz4 \mathrm { oz } of Food A;6ozA ; 6 \mathrm { oz } of Food B; 2oz2 \mathrm { oz } of Food CC
Question
Add and Subtract Matrices
Let A=[5−45−5−87]A = \left[ \begin{array} { r r } 5 & - 4 \\ 5 & - 5 \\ - 8 & 7 \end{array} \right] and B=[−76−894−4]B = \left[ \begin{array} { r r } - 7 & 6 \\ - 8 & 9 \\ 4 & - 4 \end{array} \right] . Find A+BA + B

A)
[−22−34−43]\left[ \begin{array} { l l l } - 2 & 2 \\ - 3 & 4 \\ - 4 & 3 \end{array} \right]
B)
[12−1013−14−122]\left[ \begin{array} { r r } 12 & - 10 \\ 13 & - 14 \\ - 12 & 2 \end{array} \right]
C)
[−223−5−4−3]\left[ \begin{array} { r r } - 2 & 2 \\ 3 & - 5 \\ - 4 & - 3 \end{array} \right]
D)
[−2−5−34−43]\left[ \begin{array} { r r } - 2 & - 5 \\ - 3 & 4 \\ - 4 & 3 \end{array} \right]
Question
Add and Subtract Matrices
Let A=[3−5−4]A = \left[ \begin{array} { r } 3 \\ - 5 \\ - 4 \end{array} \right] and B=[−539]B = \left[ \begin{array} { r } - 5 \\ 3 \\ 9 \end{array} \right] . Find A+BA + B .

A)
[−2−25]\left[ \begin{array} { r } - 2 \\ - 2 \\ 5 \end{array} \right]
B)
[−2−25]\left[ \begin{array} { l l l } - 2 & - 2 & 5 \end{array} \right]
C)
[3−5−53−49]\left[ \begin{array} { r r } 3 & - 5 \\ - 5 & 3 \\ - 4 & 9 \end{array} \right]
D)
[256]\left[ \begin{array} { l } 2 \\ 5 \\ 6 \end{array} \right]
Question
Solve Problems Involving Systems Without Unique Solutions
A company that manufactures products A, B, and C does both assembly and testing. The hours needed to assemble and test each product are shown in the table below.  Hours needed  weekly to assemble  Hours needed  weekly to test  Product A 14 Product B 15 Product C 210\begin{array}{l|cc} & \begin{array}{l}\text { Hours needed } \\\text { weekly to assemble }\end{array} & \begin{array}{c}\text { Hours needed } \\\text { weekly to test }\end{array} \\\hline \text { Product A } & 1 & 4 \\\text { Product B } & 1 & 5 \\\text { Product C } & 2 & 10\end{array}

The company has exactly 27 hours per week available for assembly and 120 hours per week available for testing. If the company must produce tt units of Product CC this week, how many units of Products AA and BB can they produce?

A) 15 of Product A; −2t+12- 2 t + 12 of Product B
B) 15 t of Product A; 2t+122 t + 12 of Product BB
C) t+15t + 15 of Product A;t+12A ; t + 12 of Product BB
D) 15 of Product A; 12 of Product B
Question
Solve Matrix Equations
Let A=[9449−36]\mathrm { A } = \left[ \begin{array} { r r } 9 & 4 \\ 4 & 9 \\ - 3 & 6 \end{array} \right] and B=[−2−19−9−54];X−B=A\mathrm { B } = \left[ \begin{array} { r r } - 2 & - 1 \\ 9 & - 9 \\ - 5 & 4 \end{array} \right] ; \quad \mathrm { X } - \mathrm { B } = \mathrm { A }

A)
[73130−810]\left[ \begin{array} { c c } 7 & 3 \\ 13 & 0 \\ - 8 & 10 \end{array} \right]
B)
[115−518214]\left[ \begin{array} { r r } 11 & 5 \\ - 5 & 18 \\ 2 & 14 \end{array} \right]
C)
[73−139−8−10]\left[ \begin{array} { r r } 7 & 3 \\ - 13 & 9 \\ - 8 & - 10 \end{array} \right]

D)
[79130−810]\left[ \begin{array} { r r } 7 & 9 \\ 13 & 0 \\ - 8 & 10 \end{array} \right]
Question
Multiply Matrices
A=[3−2104−3],B=[30−21]A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 3 \end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \end{array} \right]

A) AB\mathrm { AB } is not defined.
B)
[9−63−68−5]\left[\begin{array}{rrr}9 & -6 & 3 \\-6 & 8 & -5\end{array}\right]

C) [9−6−683−5]\left[\begin{array}{rr}9 & -6 \\-6 & 8 \\3 & -5\end{array}\right]
D)
[9004]\left[ \begin{array} { l l } 9 & 0 \\ 0 & 4 \end{array} \right]
Question
Solve Matrix Equations
Let A=[45039−9]\mathrm { A } = \left[ \begin{array} { c r } 4 & 5 \\ 0 & 3 \\ 9 & - 9 \end{array} \right] and B=[9−9−4508];B−X=3 A\mathrm { B } = \left[ \begin{array} { r r } 9 & - 9 \\ - 4 & 5 \\ 0 & 8 \end{array} \right] ; \quad \mathrm { B } - \mathrm { X } = 3 \mathrm {~A}

A)
X=[−3−24−4−4−2735]X = \left[ \begin{array} { c c } - 3 & - 24 \\ - 4 & - 4 \\ - 27 & 35 \end{array} \right]
B)
X=[21641427−19]X = \left[ \begin{array} { r r } 21 & 6 \\ 4 & 14 \\ 27 & - 19 \end{array} \right]
C)
X=[−3−244−4−2735]X = \left[ \begin{array} { r r } - 3 & - 24 \\ 4 & - 4 \\ - 27 & 35 \end{array} \right]
D)
X=[2164149−19]X = \left[ \begin{array} { r r } 21 & 6 \\ 4 & 14 \\ 9 & - 19 \end{array} \right]
Question
Multiply Matrices
A=[1−61−34−2],B=[−1−4−7]A = \left[ \begin{array} { r r r } 1 & - 6 & 1 \\- 3 & 4 & - 2\end{array} \right] , B = \left[ \begin{array} { l } - 1 \\- 4 \\- 7\end{array} \right]

A) [161]\left[ \begin{array} { r } 16 \\ 1 \end{array} \right]
B) AB\mathrm { AB } is not defined.
C) [ 161]\left. \begin{array} { l l } 16 & 1 \end{array} \right]
D)
[1−61−34−2−1−4−7]\left[ \begin{array} { r c r } 1 & - 6 & 1 \\- 3 & 4 & - 2 \\- 1 & - 4 & - 7\end{array} \right]
Question
Multiply Matrices
A=[−1316],B=[0−251−32]A = \left[ \begin{array} { r r } - 1 & 3 \\ 1 & 6 \end{array} \right] , B = \left[ \begin{array} { l l l } 0 & - 2 & 5 \\ 1 & - 3 & 2 \end{array} \right]


A)
[3−716−2017]\left[ \begin{array} { r r r } 3 & - 7 & 1 \\6 & - 20 & 17\end{array} \right]
B) AB\mathrm { AB } is not defined.
C)
[36−7−20117]\left[ \begin{array} { r r } 3 & 6 \\ - 7 & - 20 \\ 1 & 17 \end{array} \right]
D)
[0−6151−1812]\left[ \begin{array} { c c c } 0 & - 6 & 15 \\ 1 & - 18 & 12 \end{array} \right]
Question
Perform Scalar Multiplication
Let A=[1−32]\mathrm { A } = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right] and B=[−13−2]\mathrm { B } = \left[ \begin{array} { r } - 1 \\ 3 \\ - 2 \end{array} \right] . Find A−4 B\mathrm { A } - 4 \mathrm {~B} .

A)
[5−1510]\left[ \begin{array} { r } 5 \\ - 15 \\ 10 \end{array} \right]
B)
[−39−6]\left[ \begin{array} { r } - 3 \\ 9 \\ - 6 \end{array} \right]
C)
[−515−10]\left[ \begin{array} { r } - 5 \\ 15 \\ - 10 \end{array} \right]
D)
[5−64]\left[ \begin{array} { r } 5 \\ - 6 \\ 4 \end{array} \right]
Question
Solve Matrix Equations
Let A=[5−2−12]A = \left[ \begin{array} { r r } 5 & - 2 \\ - 1 & 2 \end{array} \right] and B=[−1−555];X+A=BB = \left[ \begin{array} { r r } - 1 & - 5 \\ 5 & 5 \end{array} \right] ; \quad X + A = B

A)
X=[−6−363]X = \left[ \begin{array} { r r } - 6 & - 3 \\ 6 & 3 \end{array} \right]
B)

X=[−3−636]X = \left[ \begin{array} { r r } - 3 & - 6 \\ 3 & 6 \end{array} \right]
C)
X=[63−6−3]X = \left[ \begin{array} { r r } 6 & 3 \\ - 6 & - 3 \end{array} \right]
D)
X=[3−6−36]X = \left[ \begin{array} { r r } 3 & - 6 \\ - 3 & 6 \end{array} \right]
Question
Multiply Matrices
A=[−65−3−7−8−5−22−7],B=[1−8−15−62−211]A = \left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]

A)
[251513−3799−1422−3−1]\left[ \begin{array} { r r r } 25 & 15 & 13 \\- 37 & 99 & - 14 \\22 & - 3 & - 1\end{array} \right]
B)
[25−37221599−313−14−1]\left[\begin{array}{rrr}25 & -37 & 22 \\15 & 99 & -3 \\13 & -14 & -1\end{array}\right]
C)

[−65−3−7−8−5−22−71−8−15−62−211]\left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7 \\1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]
D)
[−6−403−3548−1042−7]\left[\begin{array}{rrr}-6 & -40 & 3 \\-35 & 48 & -10 \\4 & 2 & -7\end{array}\right]

Question
Solve Matrix Equations
Let A=[7−7−4010−1]\mathrm { A } = \left[ \begin{array} { r r } 7 & - 7 \\ - 4 & 0 \\ 10 & - 1 \end{array} \right] and B=[1000−27−1];4X+A=B\mathrm { B } = \left[ \begin{array} { r r } 10 & 0 \\ 0 & - 2 \\ 7 & - 1 \end{array} \right] ; \quad 4 X + A = B

A)
X=[34741−12−340]X=[−34−74−112340]\mathrm { X } = \left[ \begin{array} { c c } \frac { 3 } { 4 } & \frac { 7 } { 4 } \\ 1 & - \frac { 1 } { 2 } \\ - \frac { 3 } { 4 } & 0 \end{array} \right] \quad \mathrm { X } = \left[ \begin{array} { c c } - \frac { 3 } { 4 } & - \frac { 7 } { 4 } \\ - 1 & \frac { 1 } { 2 } \\ \frac { 3 } { 4 } & 0 \end{array} \right]
B)
X=[−34−74−112340]X=\left[\begin{array}{cc}-\frac{3}{4} & -\frac{7}{4} \\-1 & \frac{1}{2} \\\frac{3}{4} & 0\end{array}\right]

C)
X=[374−2−30]X=\left[\begin{array}{rr}3 & 7 \\4 & -2 \\-3 & 0\end{array}\right]

D)
X=[−37−4230]X = \left[ \begin{array} { r r } - 3 & 7 \\ - 4 & 2 \\ 3 & 0 \end{array} \right]
Question
Model Applied Situations with Matrix Operations
Adjust the contrast by changing the black to dark grey and the light grey to white. Use matrix addition to accomplish this. A) [131131333]+[−1−1−1−1−1−1−1−1−1]=[020020222]\left[ \begin{array} { l l l } 1 & 3 & 1 \\ 1 & 3 & 1 \\ 3 & 3 & 3 \end{array} \right] + \left[ \begin{array} { l l l } - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \end{array} \right] = \left[ \begin{array} { l l l } 0 & 2 & 0 \\ 0 & 2 & 0 \\ 2 & 2 & 2 \end{array} \right]
B)
[131131333]+[0−100−10−1−1−1]=[121121222]\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { r r r } 0 & - 1 & 0 \\0 & - 1 & 0 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 1 & 2 & 1 \\1 & 2 & 1 \\2 & 2 & 2\end{array} \right]
C)
[131131333]+[111111111]=[242242444]\left[ \begin{array} { l l l } 1 & 3 & 1 \\ 1 & 3 & 1 \\ 3 & 3 & 3 \end{array} \right] + \left[ \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right] = \left[ \begin{array} { l l l } 2 & 4 & 2 \\ 2 & 4 & 2 \\ 4 & 4 & 4 \end{array} \right]
D)
[131131333]+[0−100−10−1−1−1]=[020020222]\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { r r r } 0 & - 1 & 0 \\0 & - 1 & 0 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 0 & 2 & 0 \\0 & 2 & 0 \\2 & 2 & 2\end{array} \right]
Question
Multiply Matrices
A=[−1322],B=[−20−15]A = \left[ \begin{array} { r r } - 1 & 3 \\2 & 2\end{array} \right] , B = \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 5\end{array} \right]

A)
[−115−610]\left[ \begin{array} { l l } - 1 & 15 \\ - 6 & 10 \end{array} \right]
B)
[20−210]\left[ \begin{array} { r r } 2 & 0 \\ - 2 & 10 \end{array} \right]
C)
[2−6−17]\left[ \begin{array} { r r } 2 & - 6 \\ - 1 & 7 \end{array} \right]
D)
[15−110−6]\left[ \begin{array} { l l } 15 & - 1 \\ 10 & - 6 \end{array} \right]
Question
Multiply Matrices

A)
[−68−79][ - 68 - 79 ]
B) AB\mathrm { AB } is not defined.
C)
[−68−79]\left[ \begin{array} { l } - 68 \\ - 79 \end{array} \right]
D)

[−5−8−68−3−6−239]\left[ \begin{array} { r r r } - 5 & - 8 & - 6 \\ 8 & - 3 & - 6 \\ - 2 & 3 & 9 \end{array} \right]
Question
Multiply Matrices
A=[−1−95],B=[1−59562−5−9−8]A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]

A) [−71−94−67][ - 71 - 94 - 67 ]
B)
[−71−94−67]\left[ \begin{array} { l } - 71 \\ - 94 \\ - 67 \end{array} \right]
C)
[−1−951−59562−5−9−8]\left[ \begin{array} { r r r } - 1 & - 9 & 5 \\ 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]
D)
[−14545−5−5410581−40]\left[ \begin{array} { r r r } - 1 & 45 & 45 \\ - 5 & - 54 & 10 \\ 5 & 81 & - 40 \end{array} \right]
Question
Perform Scalar Multiplication
Let A=[−32]A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right] and B=[10]B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right] . Find 2A+3B2 A + 3 B .

A) [−34][ - 34 ]
B) [−64][ - 64 ]
C) [−54]\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]
D) [02]\left[ \begin{array} { l l } 0 & 2 \end{array} \right]
Question
Multiply Matrices
A=[−628],B=[70−3]A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]

A) [−66][ - 66 ]
B) [330][ 330 ]
C) [−420−24][ - 420 - 24 ]
D)
[−420−24]\left[ \begin{array} { r } - 42 \\0 \\- 24\end{array} \right]
Question
Multiply Matrices
A=[3−2104−2],B=[50−22]A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 2 \end{array} \right] , B = \left[ \begin{array} { r r } 5 & 0 \\ - 2 & 2 \end{array} \right]

A) AB\mathrm { AB } is not defined.
B) [15−105−612−6]\left[ \begin{array} { r r r } 15 & - 10 & 5 \\ - 6 & 12 & - 6 \end{array} \right]
C)
[15−6−10125−6]\left[ \begin{array} { r r } 15 & - 6 \\- 10 & 12 \\5 & - 6\end{array} \right]
D)
[15008]\left[\begin{array}{rr}15 & 0 \\0 & 8\end{array}\right]


Question
Solve Matrix Equations
Let A=[1−1−3−1]\mathrm { A } = \left[ \begin{array} { r } 1 - 1 \\ - 3 - 1 \end{array} \right] and B=[−3−41−5];X+A=B\mathrm { B } = \left[ \begin{array} { r } - 3 - 4 \\ 1 - 5 \end{array} \right] ; \quad \mathrm { X } + \mathrm { A } = \mathrm { B }

A)
X=[−4−34−4]X = \left[ \begin{array} { r r } - 4 & - 3 \\ 4 & - 4 \end{array} \right]
B)
X=[−3−4−44]X = \left[ \begin{array} { r r } - 3 & - 4 \\ - 4 & 4 \end{array} \right]
C)
X=[4−4−4−3]X = \left[ \begin{array} { r r } 4 & - 4 \\- 4 & - 3 \end{array} \right]
D)
X=[−44−3−4]X = \left[ \begin{array} { r r } - 4 & 4 \\- 3 & - 4\end{array} \right]
Question
Solve Matrix Equations
Let A=[21−23001−22]\mathrm { A } = \left[ \begin{array} { r r r } 2 & 1 & - 2 \\ 3 & 0 & 0 \\ 1 & - 2 & 2 \end{array} \right] and B=[−1−2−2011302];2 B−2 A=X\mathrm { B } = \left[ \begin{array} { r r r } - 1 & - 2 & - 2 \\ 0 & 1 & 1 \\ 3 & 0 & 2 \end{array} \right] ; \quad 2 \mathrm {~B} - 2 \mathrm {~A} = \mathrm { X }

A)
X=[−6−60−622440]X = \left[ \begin{array} { r r r } - 6 & - 6 & 0 \\ - 6 & 2 & 2 \\ 4 & 4 & 0 \end{array} \right]
B)
X=[−622440−6−60]X = \left[ \begin{array} { r r r } - 6 & 2 & 2 \\ 4 & 4 & 0 \\ - 6 & - 6 & 0 \end{array} \right]
C)
X=[−6−60−611440]X = \left[ \begin{array} { r r r } - 6 & - 6 & 0 \\ - 6 & 1 & 1 \\ 4 & 4 & 0 \end{array} \right]
D)
X=[−611440−6−60]X = \left[ \begin{array} { r r r } - 6 & 1 & 1 \\ 4 & 4 & 0 \\ - 6 & - 6 & 0 \end{array} \right]
Question
Multiply Matrices
A=[13−1203],B=[30−1103]A = \left[ \begin{array} { r r r } 1 & 3 & - 1 \\2 & 0 & 3\end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\- 1 & 1 \\0 & 3\end{array} \right]

A)
[0069]\left[ \begin{array} { l l } 0 & 0 \\ 6 & 9 \end{array} \right]
B) ABA B is not defined.
C)
[3−30009]\left[ \begin{array} { r r r } 3 & - 3 & 0 \\ 0 & 0 & 9 \end{array} \right]
D)
[0096]\left[ \begin{array} { l l } 0 & 0 \\ 9 & 6 \end{array} \right]
Question
Perform Scalar Multiplication
Let A=[−9−38−1−68966]A = \left[ \begin{array} { r r r } - 9 & - 3 & 8 \\ - 1 & - 6 & 8 \\ 9 & 6 & 6 \end{array} \right] and B=[−389−5−41619]B = \left[ \begin{array} { r r r } - 3 & 8 & 9 \\ - 5 & - 4 & 1 \\ 6 & 1 & 9 \end{array} \right] . Find 2A - 4B.

A)
[−6−38−2018412−68−24]\left[ \begin{array} { r r r } - 6 & - 38 & - 20 \\ 18 & 4 & 12 \\ - 6 & 8 & - 24 \end{array} \right]
B)
[−21225−7−1617241321]\left[\begin{array}{rrr}-21 & 2 & 25 \\-7 & -16 & 17 \\24 & 13 & 21\end{array}\right]
C)
[−12517−6−10915715]\left[ \begin{array} { r r r } - 12 & 5 & 17 \\ - 6 & - 10 & 9 \\ 15 & 7 & 15 \end{array} \right]
D)
[−12−6155−10717915]\left[ \begin{array} { r r r } - 12 & - 6 & 15 \\ 5 & - 10 & 7 \\ 17 & 9 & 15 \end{array} \right]
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Deck 9: Matrices and Determinants
1
Use Matrices and Gaussian Elimination to Solve Systems
x+y+z=2x−y+5z=−42x+y+z=7\begin{aligned}x + y + z & = 2 \\x - y + 5 z & = - 4 \\2 x + y + z & = 7\end{aligned}

A) {(5,−1,−2)}\{ ( 5 , - 1 , - 2 ) \}
B) {(5,−2,−1)}\{ ( 5 , - 2 , - 1 ) \}
C) {(−2,−1,5)}\{ ( - 2 , - 1,5 ) \}
D) {(−2,5,−1)}\{ ( - 2,5 , - 1 ) \}
A
2
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.
[11−11−301−44000161700017]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & - 3 \\ 0 & 1 & - 4 & 4 & 0 \\ 0 & 0 & 1 & 6 & 17 \\ 0 & 0 & 0 & 1 & 7 \end{array} \right]

A) {(93,−128,−25,7)}\{ ( 93 , - 128 , - 25,7 ) \}
В) {(−3,0,17,7)}\{ ( - 3,0,17,7 ) \}
C) {(−5,−1,10,6)}\{ ( - 5 , - 1,10,6 ) \}
D) {(7,−25,−128,93)}\{ ( 7 , - 25 , - 128,93 ) \}
A
3
Perform Matrix Row Operations
[5−414−502−3−11−2−1]−5R1+R2\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }

A)
[5−414−3020−3−23−11−2−1]\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 30 & 20 & - 3 & - 23 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
B)
[5−41420−20717−11−2−1]\left[ \begin{array} { c r r | c } 5 & - 4 & 1 & 4 \\ 20 & - 20 & 7 & 17 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
C)
[30−4−919−502−3−11−2−1]\left[ \begin{array} { r r r | r } 30 & - 4 & - 9 & 19 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
D)
[−3020−3−23−502−3−11−2−1]\left[ \begin{array} { c r r | r } - 30 & 20 & - 3 & - 23 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right]
A
4
Simplify Complex Rational Expressions
9x+5z=653y+6z=272x+8y+6z=42\begin{array} { r } 9 x + 5 z = 65 \\3 y + 6 z = 27 \\2 x + 8 y + 6 z = 42\end{array}

A)
[905650362728642]\left[ \begin{array} { l l l | l } 9 & 0 & 5 & 65 \\ 0 & 3 & 6 & 27 \\ 2 & 8 & 6 & 42 \end{array} \right]

B)

[902650382756642]\left[ \begin{array} { l l | l | l } 9 & 0 & 2 & 65 \\ 0 & 3 & 8 & 27 \\ 5 & 6 & 6 & 42 \end{array} \right]
C)
[950653602728642]\left[\begin{array}{lll|l}9 & 5 & 0 & 65 \\3 & 6 & 0 & 27 \\2 & 8 & 6 & 42\end{array}\right]
D)
[905036286]\left[\begin{array}{lll}9 & 0 & 5 \\0 & 3 & 6 \\2 & 8 & 6\end{array}\right]
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5
Simplify Complex Rational Expressions
x−4y+z=15y+8z=11z=14\begin{array} { r } x - 4 y + z = 15 \\y + 8 z = 11 \\z = 14\end{array}

A)
[1−41150181100114]\left[ \begin{array} { c c c | c } 1 & - 4 & 1 & 15 \\ 0 & 1 & 8 & 11 \\ 0 & 0 & 1 & 14 \end{array} \right]
B)

[0−40150081100014]\left[ \begin{array} { c c | c | c } 0 & - 4 & 0 & 15 \\ 0 & 0 & 8 & 11 \\ 0 & 0 & 0 & 14 \end{array} \right]
C)
[141150181100114]\quadD)[1−4115118111114]\left[ \begin{array} { l l l | l } 1 & 4 & 1 & 15 \\0 & 1 & 8 & 11 \\0 & 0 & 1 & 14\end{array} \right] \quadD) \left[ \begin{array} { r r r | r } 1 & - 4 & 1 & 15 \\1 & 1 & 8 & 11 \\1 & 1 & 14\end{array} \right]
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6
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.
[653−220744202]\left[ \begin{array} { r r r | r } 6 & 5 & 3 & - 2 \\ 2 & 0 & 7 & 4 \\ 4 & 2 & 0 & 2 \end{array} \right]

A) 6x+5y+3z=−26 x + 5 y + 3 z = - 2
2x+7z=42 x + 7 z = 4
4x+2y=24 x + 2 y = 2
B) 6x−5y+3z=−26 x - 5 y + 3 z = - 2
2x+7z=−42 x + 7 z = - 4
4x+2y=−24 x + 2 y = - 2
C) 6x+5y+3z=−26 x + 5 y + 3 z = - 2
2x+7z=42 x + 7 z = 4
4x+2z=24 x + 2 z = 2
D) 6x+5y+3z=−26 x + 5 y + 3 z = - 2
2x+y+7z=42 x + y + 7 z = 4
4x+2y+z=24 x + 2 y + z = 2
4x+2y+z=24 x + 2 y + z = 2
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7
Use Matrices and Gauss-Jordan Elimination to Solve Systems
x=−1−y−zx−y+4z=−63x+y=7−z\begin{array} { l } x = - 1 - y - z \\x - y + 4 z = - 6 \\3 x + y = 7 - z\end{array}

A) {(4,−2,−3)}\{ ( 4 , - 2 , - 3 ) \}
В) {(−2,−3,4)}\{ ( - 2 , - 3,4 ) \}
C) {(−3,−2,4)}\{ ( - 3 , - 2,4 ) \}
D) {(−3,4,−2)}\{ ( - 3,4 , - 2 ) \}
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8
Use Matrices and Gaussian Elimination to Solve Systems
x+y+z−w=62x−y+3z+4w=−44x+2y−z−w=−13−x−2y+4z+3w=12\begin{array} { l r } x + y + z - w = & 6 \\2 x - y + 3 z + 4 w = & - 4 \\4 x + 2 y - z - w = & - 13 \\- x - 2 y + 4 z + 3 w = & 12\end{array}

A) {(−4,3,5,−2)}\{ ( - 4,3,5 , - 2 ) \}
B) {(4,−3,−5,2)}\{ ( 4 , - 3 , - 5,2 ) \}
C) {(−14,13,15,−12)}\left\{ \left( - \frac { 1 } { 4 } , \frac { 1 } { 3 } , \frac { 1 } { 5 } , - \frac { 1 } { 2 } \right) \right\}
D) {(14,−13,−15,12)}\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 3 } , - \frac { 1 } { 5 } , \frac { 1 } { 2 } \right) \right\}
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9
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.
[15−8401−6−4001−7]\left[ \begin{array} { r r r | r } 1 & 5 & - 8 & 4 \\ 0 & 1 & - 6 & - 4 \\ 0 & 0 & 1 & - 7 \end{array} \right]

A) {(178,−46,−7)}\{ ( 178 , - 46 , - 7 ) \}
B) {(4,−4,−7)}\{ ( 4 , - 4 , - 7 ) \}
C) {(6,1,−8)}\{ ( 6,1 , - 8 ) \}
D) {(250,38,−7)}\{ ( 250,38 , - 7 ) \}
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10
Use Matrices and Gauss-Jordan Elimination to Solve Systems
−4x+7y−z=39x+6y+9z=62−5x+y+z=−1\begin{array} { r } - 4 x + 7 y - z = 39 \\x + 6 y + 9 z = 62 \\- 5 x + y + z = - 1\end{array}

A) {(2,7,2)}\{ ( 2,7,2 ) \}
B) {(2,2,7)}\{ ( 2,2,7 ) \}
C) {(−2,7,4)}\{ ( - 2,7,4 ) \}
D) {(4,7,−2)}\{ ( 4,7 , - 2 ) \}
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11
Perform Matrix Row Operations
[−20601045113−302−7421]15R1\left[ \begin{array} { r r r | r } - 20 & 60 & 10 & 45 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right] \frac { 1 } { 5 } \mathrm { R } _ { 1 }

A)
[−41229113−302−7421]\left[ \begin{array} { c c r | r } - 4 & 12 & 2 & 9 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right]
B)

[−2060104515135−3502−7421]\left[ \begin{array} { c c r | r } - 20 & 60 & 10 & 45 \\\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \\2 & - 7 & 4 & 21\end{array} \right]
C)
[−412245113−302−7421]\quadD)[−4122915135−35025−7545215]\left[ \begin{array} { r r r | r } - 4 & 12 & 2 & 45 \\1 & 13 & - 3 & 0 \\2 & - 7 & 4 & 21\end{array} \right] \quadD) \left[ \begin{array} { r r r | r } - 4 & 12 & 2 & 9 \\\frac { 1 } { 5 } & \frac { 13 } { 5 } & - \frac { 3 } { 5 } & 0 \\\frac { 2 } { 5 } - \frac { 7 } { 5 } & \frac { 4 } { 5 } & \frac { 21 } { 5 }\end{array} \right]
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12
Use Matrices and Gauss-Jordan Elimination to Solve Systems
−3x−y−7z=−763x+3y−6z=−27−9x−2y+z=−23\begin{aligned}- 3 x - y - 7 z & = - 76 \\3 x + 3 y - 6 z & = - 27 \\- 9 x - 2 y + z & = - 23\end{aligned}

A) {(2,7,9)}\{ ( 2,7,9 ) \}
B) {(2,9,7)}\{ ( 2,9,7 ) \}
C) {(−2,7,4)}\{ ( - 2,7,4 ) \}
D) {(4,7,−2)}\{ ( 4,7 , - 2 ) \}
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13
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. Then use back-substitution to find the solution.
[13211201−12−80014]\left[ \begin{array} { r r r | r } 1 & \frac { 3 } { 2 } & 1 & \frac { 1 } { 2 } \\ 0 & 1 & - \frac { 1 } { 2 } & - 8 \\ 0 & 0 & 1 & 4 \end{array} \right]

A) {(112,−6,4)}\left\{ \left( \frac { 11 } { 2 } , - 6,4 \right) \right\}
B) {(12,−8,4)}\left\{ \left( \frac { 1 } { 2 } , - 8,4 \right) \right\}
C) {(−3,−172,3)}\left\{ \left( - 3 , - \frac { 17 } { 2 } , 3 \right) \right\}
D) {(−212,−6,4)}\left\{ \left( - \frac { 21 } { 2 } , - 6,4 \right) \right\}
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14
Simplify Complex Rational Expressions
12x+7y−8z+w=15y+z=−6x−y−2z=−53x−3y+4z=5\begin{array} { r } 12 x + 7 y - 8 z + w = 1 \\5 y + z = - 6 \\x - y - 2 z = - 5 \\3 x - 3 y + 4 z = 5\end{array}

A)
[127−8110510−61−1−20−53−3405]\left[ \begin{array} { l r r r | r } 12 & 7 & - 8 & 1 & 1 \\0 & 5 & 1 & 0 & - 6 \\1 & - 1 & - 2 & 0 & - 5 \\3 & - 3 & 4 & 0 & 5\end{array} \right]
B)
[127−8105161−1−2−53−345]\left[ \begin{array} { l r r | r } 12 & 7 & - 8 & 1 \\0 & 5 & 1 & 6 \\1 & - 1 & - 2 & - 5 \\3 & - 3 & 4 & 5\end{array} \right]
C)
[1201975−1−3−81−2410001−6−55]\left[ \begin{array} { l r r | r } 12 & 0 & 1 & 9 \\7 & 5 & - 1 & - 3 \\- 8 & 1 & - 2 & 4 \\1 & 0 & 0 & 0 \\1 & - 6 & - 5 & 5\end{array} \right]
D)
[1278110510−61120−533405]\left[ \begin{array} { l l l l | r } 12 & 7 & 8 & 1 & 1 \\0 & 5 & 1 & 0 & - 6 \\1 & 1 & 2 & 0 & - 5 \\3 & 3 & 4 & 0 & 5\end{array} \right]
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15
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables.
[510212−1810−69006−11030−3−4]\left[ \begin{array} { r r r r | r } 5 & 1 & 0 & 2 & 12 \\- 1 & 8 & 1 & 0 & - 6 \\9 & 0 & 0 & 6 & - 11 \\0 & 3 & 0 & - 3 & - 4\end{array} \right]

A)
5x+y+2w=125 x + y + 2 w = 12
−x+8y+z=−6- x + 8 y + z = - 6
9x+6w=−119 x + 6 w = - 11
3y−3w=−43 y - 3 w = - 4
B)
5x+y+z+2w=125 x + y + z + 2 w = 12
−x+8y+z+w=−6- x + 8 y + z + w = - 6
9x+y+z+6w=−119 x + y + z + 6 w = - 11
x+3y+z−3w=−4x + 3 y + z - 3 w = - 4
C)
5x+y+2w=125 x + y + 2 w = 12
x+8y+z=−6x + 8 y + z = - 6
9x+6w=−119 x + 6 w = - 11
3y+3w=−43 y + 3 w = - 4
D)
5x+y+2z=125 x + y + 2 z = 12
−x+8y+z=−6- x + 8 y + z = - 6
9x+6y=−119 x + 6 y = - 11
3x−3y=−43 x - 3 y = - 4
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16
Use Matrices and Gaussian Elimination to Solve Systems
x−y+4z=152x+z=3x+2y+z=−3\begin{aligned}x - y + 4 z & = 15 \\2 x + z & = 3 \\x + 2 y + z & = - 3\end{aligned}

A) {(0,−3,3)}\{ ( 0 , - 3,3 ) \}
B) {(0,3,−3)}\{ ( 0,3 , - 3 ) \}
C) {(3,−3,0)}\{ ( 3 , - 3,0 ) \}
D) {(3,0,−3)}\{ ( 3,0 , - 3 ) \}
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17
Perform Matrix Row Operations
[11−1130−43−5030−1−41−3301−4]−2R1+R33R1+R4\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 3 & 0 & - 1 & - 4 & 1 \\ - 3 & 3 & 0 & 1 & - 4 \end{array} \right] \quad \begin{array} { r } - 2 R _ { 1 } + R _ { 3 } \\ 3 R _ { 1 } + R _ { 4 } \end{array}

A)
[11−1130−43−501−21−6−506−345]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 1 & - 2 & 1 & - 6 & - 5 \\ 0 & 6 & - 3 & 4 & 5 \end{array} \right]
B)
[11−1130−43−501−21−6−5−3301−4]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 1 & - 2 & 1 & - 6 & - 5 \\ - 3 & 3 & 0 & 1 & - 4 \end{array} \right]
C)
[11−1130−43−5052−3−2706−345]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 5 & 2 & - 3 & - 2 & 7 \\ 0 & 6 & - 3 & 4 & 5 \end{array} \right]
D)
[11−1130−43−501−21−6−106−344]\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & 3 \\ 0 & - 4 & 3 & - 5 & 0 \\ 1 & - 2 & 1 & - 6 & - 1 \\ 0 & 6 & - 3 & 4 & 4 \end{array} \right]
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18
Use Matrices and Gaussian Elimination to Solve Systems
5x−y−3z=−83x−8z=−217y+z=31\begin{aligned}5 x - y - 3 z & = - 8 \\3 x - 8 z & = - 21 \\7 y + z & = 31\end{aligned}

A) {(1,4,3)}\{ ( 1,4,3 ) \}
B) {(1,3,4)}\{ ( 1,3,4 ) \}
C) {(−1,4,2)}\{ ( - 1,4,2 ) \}
D) {(−1,2,4)}\{ ( - 1,2,4 ) \}
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19
Simplify Complex Rational Expressions
2x+6y+9z=667x+6y+8z=779x+8y−2z=51\begin{array} { l } 2 x + 6 y + 9 z = 66 \\7 x + 6 y + 8 z = 77 \\9 x + 8 y - 2 z = 51\end{array}

A)
[269667687798−251]\left[ \begin{array} { r r r | r } 2 & 6 & 9 & 66 \\ 7 & 6 & 8 & 77 \\ 9 & 8 & - 2 & 51 \end{array} \right]

B)

[279666687798−251]\left[ \begin{array} { r r r | r } 2 & 7 & 9 & 66 \\ 6 & 6 & 8 & 77 \\ 9 & 8 & - 2 & 51 \end{array} \right]
C)
[669627786751−289]\left[\begin{array}{rrr|r}66 & 9 & 6 & 2 \\77 & 8 & 6 & 7 \\51 & -2 & 8 & 9\end{array}\right]

D)
[26976898−2]\left[\begin{array}{rrr}2 & 6 & 9 \\7 & 6 & 8 \\9 & 8 & -2\end{array}\right]
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20
Use Matrices and Gaussian Elimination to Solve Systems
3x+5y−2w=−132x+7z−w=−14y+3z+3w=1−x+2y+4z=−5\begin{aligned}3 x + 5 y - 2 w = & - 13 \\2 x + 7 z - w = & - 1 \\4 y + 3 z + 3 w = & 1 \\- x + 2 y + 4 z = & - 5\end{aligned}

A) {(1,−2,0,3)}\{ ( 1 , - 2,0,3 ) \}
B) {(43,−1320,0,52)}\left\{ \left( \frac { 4 } { 3 } , - \frac { 13 } { 20 } , 0 , \frac { 5 } { 2 } \right) \right\}
C) {(34,−2,0,34)}\left\{ \left( \frac { 3 } { 4 } , - 2,0 , \frac { 3 } { 4 } \right) \right\}
D) {(−1,−2013,0,25)}\left\{ \left( - 1 , - \frac { 20 } { 13 } , 0 , \frac { 2 } { 5 } \right) \right\}
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21
Use Matrices and Gauss-Jordan Elimination to Solve Systems
x+y−z+w=−53x−y+3z−2w=7−2x+2y+z−w=16−x−2y−3z+3w=−22\begin{array} { c } x + y - z + w = - 5 \\3 x - y + 3 z - 2 w = 7 \\- 2 x + 2 y + z - w = 16 \\- x - 2 y - 3 z + 3 w = - 22\end{array}

A) {(−2,3,4,−2)}\{ ( - 2,3,4 , - 2 ) \}
B) {(−2,−3,5,12)}\left\{ \left( - 2 , - 3,5 , \frac { 1 } { 2 } \right) \right\}
C) {(2,−3,−4,−2)}\{ ( 2 , - 3 , - 4 , - 2 ) \}
D) {(12,−13,−14,−12)}\left\{ \left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } , - \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right) \right\}
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22
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z+w=73x−2z+5w=11−4x+3y+w=4−x−y−z−w=6\begin{aligned}x + y + z + w & = 7 \\3 x - 2 z + 5 w & = 11 \\- 4 x + 3 y + w & = 4 \\- x - y - z - w & = 6\end{aligned}

A) ∅\varnothing
B) {(32,1,13,−2)}\left\{ \left( \frac { 3 } { 2 } , 1 , \frac { 1 } { 3 } , - 2 \right) \right\}
C) {(74,−12,5,−16)}\left\{ \left( \frac { 7 } { 4 } , - \frac { 1 } { 2 } , 5 , - \frac { 1 } { 6 } \right) \right\}
D) {(−11,719,619,−4)}\left\{ \left( - 11 , \frac { 7 } { 19 } , \frac { 6 } { 19 } , - 4 \right) \right\}
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23
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
4x−y+3z=12x+4y+6z=−325x+3y+9z=20\begin{aligned}4 x - y + 3 z & = 12 \\x + 4 y + 6 z & = - 32 \\5 x + 3 y + 9 z & = 20\end{aligned}

A) ∅\varnothing
В) {(2,−7,−1)}\{ ( 2 , - 7 , - 1 ) \}
C) {(8,−7,−2)}\{ ( 8 , - 7 , - 2 ) \}
D) {(−8,−7,9)}\{ ( - 8 , - 7,9 ) \}
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24
Apply Gaussian Elimination to Systems with More Variables than Equations
x+y+z=92x−3y+4z=7\begin{array} { l } x + y + z = 9 \\2 x - 3 y + 4 z = 7\end{array}

A) {(−75z+345,25z+115,z)}\left\{ \left( - \frac { 7 } { 5 } \mathrm { z } + \frac { 34 } { 5 } , \frac { 2 } { 5 } \mathrm { z } + \frac { 11 } { 5 } , \mathrm { z } \right) \right\}
B) {(35z+165,−85z+295,z)}\left\{ \left( \frac { 3 } { 5 } z + \frac { 16 } { 5 } , - \frac { 8 } { 5 } z + \frac { 29 } { 5 } , z \right) \right\}
C) {(275,135,1)}\left\{ \left( \frac { 27 } { 5 } , \frac { 13 } { 5 } , 1 \right) \right\}
D) ∅\varnothing
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25
Write a system of linear equations in three variables, and then use matrices to solve the system.
Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake to 103 g protein, 93 g fat, and 135 g carbohydrate. According to the health conscious hostess, the marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g carbohydrate. How many of each snack can he eat to obtain his goal?

A) 7 mushrooms; 4 meatballs; 2 eggs
B) 4 mushrooms; 2 meatballs; 7 eggs
C) 2 mushrooms; 7 meatballs; 4 eggs
D) 8 mushrooms; 5 meatballs; 3 eggs
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26
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
5x+2y+z=−112x−3y−z=177x−y=12\begin{aligned}5 x + 2 y + z & = - 11 \\2 x - 3 y - z & = 17 \\7 x - y & = 12\end{aligned}

A) ∅\varnothing
B) {(0,−6,1)}\{ ( 0 , - 6,1 ) \}
C) {(−2,0,−1)}\{ ( - 2,0 , - 1 ) \}
D) {(1,−5,0)}\{ ( 1 , - 5,0 ) \}
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27
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
3x−2y+2z−w=24x+y+z+6w=8−3x+2y−2z+w=55x+3z−2w=1\begin{array} { r } 3 x - 2 y + 2 z - w = 2 \\4 x + y + z + 6 w = 8 \\- 3 x + 2 y - 2 z + w = 5 \\5 x + 3 z - 2 w = 1\end{array}

A) ∅\varnothing
B) {(2,0,−337,937)}\left\{ \left( 2,0 , - \frac { 3 } { 37 } , \frac { 9 } { 37 } \right) \right\}
C) {(12,0,−373,379)}\left\{ \left( \frac { 1 } { 2 } , 0 , - \frac { 37 } { 3 } , \frac { 37 } { 9 } \right) \right\}
D) {(1,−13,49,6)}\left\{ \left( 1 , - \frac { 1 } { 3 } , \frac { 4 } { 9 } , 6 \right) \right\}
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28
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+3y+2z=114y+9z=−12x+7y+11z=−1\begin{array} { r } x + 3 y + 2 z = 11 \\4 y + 9 z = - 12 \\x + 7 y + 11 z = - 1\end{array}

A) {(19z4+20,−9z4−3,z)}\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } - 3 , z \right) \right\}
B) {(19z4+20,−9z4+3,z)}\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}
C) {(19z4+20,9z4+3,z)}\left\{ \left( \frac { 19 z } { 4 } + 20 , \frac { 9 z } { 4 } + 3 , z \right) \right\}
D) {(−19z4+20,−9z4+3,z)}\left\{ \left( - \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}
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29
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+8y+8z=87x+7y+z=18x+15y+9z=−9\begin{aligned}x + 8 y + 8 z & = 8 \\7 x + 7 y + z & = 1 \\8 x + 15 y + 9 z & = - 9\end{aligned}

A) ∅\varnothing
B) {(0,0,1)}\{ ( 0,0,1 ) \}
C) {(1,−1,1)}\{ ( 1 , - 1,1 ) \}
D) {(−1,0,1)}\{ ( - 1,0,1 ) \}
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30
Write a system of linear equations in three variables, and then use matrices to solve the system.
A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A tree takes 14 hours to prepare, 3 hours to paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 15 hours to paint, and 7 hours to fire. If the workshop has 85 hours for prep time, 56 hours for painting, and 100 hours for firing, how many of each can be made?

A) 7 wreaths; 4 trees; 2 sleighs
B) 4 wreaths; 2 trees; 7 sleighs
C) 2 wreaths; 7 trees; 4 sleighs
D) 8 wreaths; 5 trees; 3 sleighs
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31
Write a system of linear equations in three variables, and then use matrices to solve the system.
There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sales by age group for marketing purposes. The percentage of 36- to 59-year-old buyers and the percentage of buyers 60 and older combined exceeds the percentage of buyers 35 and younger by 40%. If the percentage of buyers in the oldest group is doubled, it is 34% less than the percentage of users in the middle group. Find the percentage of buyers in each of the three age groups.

A) 30% 35 and younger; 58% 36-59 year olds; 12% 60 and older
B) 32% 35 and younger; 55% 36-59 year olds; 13% 60 and older
C) 24% 35 and younger; 60% 36-59 year olds; 16% 60 and older
D) 12% 35 and younger; 58% 36-59 year olds; 30% 60 and older
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32
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z=92x−3y+4z=7x−4y+3z=−2\begin{array} { r r } x + y + z & = 9 \\2 x - 3 y + 4 z & = 7 \\x - 4 y + 3 z & = - 2\end{array}

A) {(−7z5+345,2z5+115,z)}\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}
B) {(z5+345,2z5+115,z)}\left\{ \left( \frac { z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}
C) {(−7z5+345,2z5−115,z)}\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}
D) {(7z5+345,2z5−115,z)}\left\{ \left( \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , \mathrm { z } \right) \right\}
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33
Apply Gaussian Elimination to Systems with More Variables than Equations
x+y+z=7x−y+2z=7\begin{array} { l } x + y + z = 7 \\x - y + 2 z = 7\end{array}

A) {(−32z+7,12z,z)}\left\{ \left( - \frac { 3 } { 2 } z + 7 , \frac { 1 } { 2 } z , z \right) \right\}
B) {(−3z+14,2z−7,z)}\{ ( - 3 z + 14,2 z - 7 , z ) \}
C) {(4,1,2)}\{ ( 4,1,2 ) \}
D) {(8,−3,2)}\{ ( 8 , - 3,2 ) \}
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34
Apply Gaussian Elimination to Systems with More Variables than Equations
3x+y+z−2w=102x+3y+3z+w=−52x+y+4z+11w=11\begin{array} { r } 3 x + y + z - 2 w = 10 \\2 x + 3 y + 3 z + w = - 5 \\2 x + y + 4 z + 11 w = 11\end{array}

A) {(w+5,3w−7,−4w+2,w)}\{ ( w + 5,3 w - 7 , - 4 w + 2 , w ) \}
B) {(2w+3,6w−7,−10w+8,w)}\{ ( 2 w + 3,6 w - 7 , - 10 w + 8 , w ) \}
C) {(6,−4,−2,1)}\{ ( 6 , - 4 , - 2,1 ) \}
D) {(7,−1,−6,2)}\{ ( 7 , - 1 , - 6,2 ) \}
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35
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z+w=83x+2y+z+4w=214x+4y+5z+8w=302x+3y+6z+9w=15\begin{array} { r } x + y + z + w = 8 \\3 x + 2 y + z + 4 w = 21 \\4 x + 4 y + 5 z + 8 w = 30 \\2 x + 3 y + 6 z + 9 w = 15\end{array}

A) {(−6w+3,9w+7,−4w−2,w)}\{ ( - 6 w + 3,9 w + 7 , - 4 w - 2 , w ) \}
B) {(5w+11,−3w−7,−3w+4,w)}\{ ( 5 w + 11 , - 3 w - 7 , - 3 w + 4 , w ) \}
C) {(−3,16,−6,1)}\{ ( - 3,16 , - 6,1 ) \}
D) ∅\varnothing
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36
Apply Gaussian Elimination to Systems with More Variables than Equations
5x−y+z=87x+y+z=6\begin{array} { l } 5 x - y + z = 8 \\7 x + y + z = 6\end{array}

A) {(−16z+76,16z−136,z)}\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}
B) {(−z+3,4z+7,z)}\{ ( - z + 3,4 z + 7 , z ) \}
C) {(16z+76,16z,z)}\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}
D) ∅\varnothing
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37
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x−y+z−w=10−2x+3y+5w=−28x+2y+8z+3w=−10x−4y−6z−5w=30\begin{array} { r } x - y + z - w = 10 \\- 2 x + 3 y + 5 w = - 28 \\x + 2 y + 8 z + 3 w = - 10 \\x - 4 y - 6 z - 5 w = 30\end{array}

A) {(−17w−10,−13w−16,5w+4,w)}\{ ( - 17 w - 10 , - 13 w - 16,5 w + 4 , w ) \}
B) {(3w−2,−8w+3,4w+9,w)}\{ ( 3 w - 2 , - 8 w + 3,4 w + 9 , w ) \}
C) {(24,10,−6,−2)}\{ ( 24,10 , - 6 , - 2 ) \}
D) ∅\varnothing
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38
Use Matrices and Gauss-Jordan Elimination to Solve Systems
3x+5y+2w=−122x+6z−w=−5−2y+3z−3w=−3−x+2y+4z+w=−2\begin{aligned}3 x + 5 y + 2 w & = - 12 \\2 x + 6 z - w & = - 5 \\- 2 y + 3 z - 3 w & = - 3 \\- x + 2 y + 4 z + w & = - 2\end{aligned}

A) {(−1,−3,0,3)}\{ ( - 1 , - 3,0,3 ) \}
B) {(1,−3,0,3)}\{ ( 1 , - 3,0,3 ) \}
C) {(−1,3,0,−3)}\{ ( - 1,3,0 , - 3 ) \}
D) {(1,3,0,−3)}\{ ( 1,3,0 , - 3 ) \} -
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39
Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
x+y+z=7x−y+2z=72x+3z=14\begin{array} { r } x + y + z = 7 \\x - y + 2 z = 7 \\2 x + 3 z = 14\end{array}

A) {(−3z2+7,z2,z)}\left\{ \left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right) \right\}
B) {(−3z2−7,z2,z)}\left\{ \left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right) \right\}
C) {(−3z2+7,2z,z)}\left\{ \left( - \frac { 3 z } { 2 } + 7,2 z , z \right) \right\}
D) {(−3z2−7,2z,z)}\left\{ \left( - \frac { 3 z } { 2 } - 7,2 z , z \right) \right\}
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40
Write a system of linear equations in three variables, and then use matrices to solve the system.
The table below shows the number of birds for three selected years after an endangered species protection program was started. x (Number of years after 1980) 1510y (Number of birds) 43139349\begin{array} { l | c c c } \mathrm { x } \text { (Number of years after 1980) } & 1 & 5 & 10 \\\hline \mathrm { y } \text { (Number of birds) } & 43 & 139 & 349\end{array} Use the quadratic function y=ax2+bx+cy = a x ^ { 2 } + b x + c to model the data. Solve the system of linear equations involving a,ba , b , and cc using matrices. Find the equation that models the data.

A) y=2x2+12x+29y = 2 x ^ { 2 } + 12 x + 29
B) y=3x2+24x+24y = 3 x ^ { 2 } + 24 x + 24
C) y=4x2−12x+32y = 4 x ^ { 2 } - 12 x + 32
D) y=4x2−36x+25y = 4 x ^ { 2 } - 36 x + 25
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41
Perform Scalar Multiplication
Let A=[−3602]\mathrm { A } = \left[ \begin{array} { r r } - 3 & 6 \\ 0 & 2 \end{array} \right] . Find 4A.

A)
[−122408]\left[ \begin{array} { r r } - 12 & 24 \\0 & 8\end{array} \right]
B)
[−122402]\left[\begin{array}{rr}-12 & 24 \\0 & 2\end{array}\right]
C)
[−12602]\left[\begin{array}{rr}-12 & 6 \\0 & 2\end{array}\right]
D)
[11046]\left[\begin{array}{rr}1 & 10 \\4 & 6\end{array}\right]
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42
Add and Subtract Matrices
Let A=[−1031]\mathrm { A } = \left[ \begin{array} { r r } - 1 & 0 \\ 3 & 1 \end{array} \right] and B=[−1331]\mathrm { B } = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 1 \end{array} \right] . Find A−B\mathrm { A } - \mathrm { B } .

A)
[0−300]\left[ \begin{array} { r r } 0 & - 3 \\ 0 & 0 \end{array} \right]
B)
[−2362]\left[ \begin{array} { r } - 23 \\ 62 \end{array} \right]
C)
[0300]\left[ \begin{array} { l l } 0 & 3 \\ 0 & 0 \end{array} \right]
D)
[−3][ - 3 ]
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43
Understand What is Meant by Equal Matrices
[x1]=[−1y]\left[ \begin{array} { c } x \\1\end{array} \right] = \left[ \begin{array} { l } - 1 \\y\end{array} \right]

A) x=−1;y=1x = - 1 ; y = 1
B) x=1;y=−1x = 1 ; y = - 1
C) x=−1;y=−1x = - 1 ; y = - 1
D) x=1;y=1x = 1 ; y = 1
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44
Understand What is Meant by Equal Matrices
[x+3y+475]=[7−37z]\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 5\end{array} \right] = \left[ \begin{array} { r r } 7 & - 3 \\7 & z\end{array} \right]
B) x=−4;y=7;z=−5x = - 4 ; y = 7 ; z = - 5

A) x=4;y=−7;z=5x = 4 ; y = - 7 ; z = 5
D) x=4;y=5;z=7x = 4 ; y = 5 ; z = 7
C) x=7;y=−3;z=5x = 7 ; y = - 3 ; z = 5
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45
Understand What is Meant by Equal Matrices
[xy+69z3]=[88903]\left[ \begin{array} { r r } x & y + 6 \\9 z & 3\end{array} \right] = \left[ \begin{array} { c c } 8 & 8 \\90 & 3\end{array} \right]

A) x=8;y=2;z=10x = 8 ; y = 2 ; z = 10
B) x=8;y=8;z=90x = 8 ; y = 8 ; z = 90
C) x=8;y=3;z=8x = 8 ; y = 3 ; z = 8
D) x=3;y=14;z=810x = 3 ; y = 14 ; z = 810
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46
Matrix Operations and Their Applications
1 Use Matrix Notation
[3767−33−e−15−13π−69−121281396−15];a34\left[ \begin{array} { c c c c c } 3 & 7 & 6 & 7 & - 3 \\3 & - e & - 15 & - 13 & \pi \\- 6 & 9 & - 12 & 12 & 8 \\\frac { 1 } { 3 } & 9 & 6 & - 1 & 5\end{array} \right] ; a _ { 34 }

A) 4×5;124 \times 5 ; 12
B) 5×4;65 \times 4 ; 6
C) 20;820 ; 8
D) 4×4;−124 \times 4 ; - 12
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47
Apply Gaussian Elimination to Systems with More Variables than Equations
2x+y+2z−4w=10x+3y+2z−11w=173x+y+7z−21w=0\begin{aligned}2 x + y + 2 z - 4 w & = 10 \\x + 3 y + 2 z - 11 w & = 17 \\3 x + y + 7 z - 21 w & = 0\end{aligned}

A) {(−3w+5,2w+6,4w−3,w)}\{ ( - 3 w + 5,2 w + 6,4 w - 3 , w ) \}
B) {(3w+5,6w+6,−4w−3,w)}\{ ( 3 w + 5,6 w + 6 , - 4 w - 3 , w ) \}
C) {(w+5,8w+4,−3w−2,w)}\{ ( w + 5,8 w + 4 , - 3 w - 2 , w ) \}
D) {(w−5,8w−4,−3w+2,w)}\{ ( w - 5,8 w - 4 , - 3 w + 2 , w ) \}
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48
Perform Scalar Multiplication
Let B=[−166−3]B = \left[ \begin{array} { l l l l } - 1 & 6 & 6 & - 3 \end{array} \right] . Find −3B- 3 B .

A) [3−18−18\left[ \begin{array} { l l l } 3 & - 18 & - 18 \end{array} \right. ]
B) [366\left[ \begin{array} { l l l } 3 & 6 & 6 \end{array} \right. ]
C) [−31818−9][ - 3 18 18 - 9 ]
D) [−344−5][ - 3 44 - 5 ]
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49
Perform Scalar Multiplication
Let A=[3324]\mathrm { A } = \left[ \begin{array} { l l } 3 & 3 \\ 2 & 4 \end{array} \right] and B=[04−16]\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right] . Find 3 A+B3 \mathrm {~A} + \mathrm { B } .

A)
[913518]\left[ \begin{array} { l l } 9 & 13 \\ 5 & 18 \end{array} \right]
B)
[921330]\left[ \begin{array} { l l } 9 & 21 \\ 3 & 30 \end{array} \right]
C)
[913110]\left[ \begin{array} { l l } 9 & 13 \\ 1 & 10 \end{array} \right]
D)
[97510]\left[ \begin{array} { c c } 9 & 7 \\ 5 & 10 \end{array} \right]
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50
Add and Subtract Matrices
Let A=[24142−1−123]\mathrm { A } = \left[ \begin{array} { r r r } 2 & 4 & 1 \\ 4 & 2 & - 1 \\ - 1 & 2 & 3 \end{array} \right] and B=[3−41−10203−2]\mathrm { B } = \left[ \begin{array} { r r r } 3 & - 4 & 1 \\ - 1 & 0 & 2 \\ 0 & 3 & - 2 \end{array} \right] . Find A+B\mathrm { A } + \mathrm { B } .

A)
[502321−151]\left[ \begin{array} { r r r } 5 & 0 & 2 \\ 3 & 2 & 1 \\ - 1 & 5 & 1 \end{array} \right]
B)
[3−82521−151]\left[ \begin{array} { r r r } 3 & - 8 & 2 \\ 5 & 2 & 1 \\ - 1 & 5 & 1 \end{array} \right]
C)
[3−8232−3151]\left[ \begin{array} { r r r } 3 & - 8 & 2 \\ 3 & 2 & - 3 \\ 1 & 5 & 1 \end{array} \right]
D)
[50252−3151]\left[ \begin{array} { r r r } 5 & 0 & 2 \\ 5 & 2 & - 3 \\ 1 & 5 & 1 \end{array} \right]
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51
Add and Subtract Matrices
Let A=[−5125]\mathrm { A } = \left[ \begin{array} { r r } - 5 & 1 \\ 2 & 5 \end{array} \right] and B=[624−2]\mathrm { B } = \left[ \begin{array} { r r } 6 & 2 \\ 4 & - 2 \end{array} \right] . Find A+B\mathrm { A } + \mathrm { B } .

A)
[1363]\left[\begin{array}{ll}1 & 3 \\6 & 3\end{array}\right]
B)
[34−13]\left[\begin{array}{r}34 \\-13\end{array}\right]
C)
[1−3−1−7]\left[\begin{array}{rr}1 & -3 \\-1 & -7\end{array}\right]

D)
[13][ 13 ]
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52
Add and Subtract Matrices
Let A=[4−1028−12−3459]\mathrm { A } = \left[ \begin{array} { r r r } 4 & - 10 & 2 \\ 8 & - 12 & - 3 \\ 4 & 5 & 9 \end{array} \right] and B=[010340−5−58−5]\mathrm { B } = \left[ \begin{array} { r r r } 0 & 10 & 3 \\ 4 & 0 & - 5 \\ - 5 & 8 & - 5 \end{array} \right] . Find A−B\mathrm { A } - \mathrm { B }

A)
[4−20−14−1229−314]\left[ \begin{array} { r r r } 4 & - 20 & - 1 \\ 4 & - 12 & 2 \\ 9 & - 3 & 14 \end{array} \right]
B)
[40−14−1221−34]\left[ \begin{array} { r r r } 4 & 0 & - 1 \\ 4 & - 12 & 2 \\ 1 & - 3 & 4 \end{array} \right]
C)
[420−14−1229314]\left[ \begin{array} { r r r } 4 & 20 & - 1 \\ 4 & - 12 & 2 \\ 9 & 3 & 14 \end{array} \right]
D)
[40512−12−8−1134]\left[ \begin{array} { r r r } 4 & 0 & 5 \\ 12 & - 12 & - 8 \\ - 1 & 13 & 4 \end{array} \right]
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53
Add and Subtract Matrices
A)
[−81−1705−6]\left[ \begin{array} { r r } - 8 & 1 \\- 17 & 0 \\5 & - 6\end{array} \right]

A)
[−81−1785−6]\left[ \begin{array} { r r } -8 & 1 \\ -17 & 8\\ 5 & -6 \end{array} \right]

B)
[1478111]\left[ \begin{array} { r r } 1 & 4 \\ 7 & 8 \\ 11 & 1 \end{array} \right]
C)
[11705−2]\left[ \begin{array} { r r } 1 & 1 \\ 7 & 0 \\ 5 & - 2 \end{array} \right]
D)
[3−270−56]\left[ \begin{array} { r r } 3 & - 2 \\ 7 & 0 \\ - 5 & 6 \end{array} \right]
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54
Solve Problems Involving Systems Without Unique Solutions
The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If construction limits z to t cars per minute, how many cars per minute must pass through the other intersections to keep traffic moving? <strong>Solve Problems Involving Systems Without Unique Solutions The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If construction limits z to t cars per minute, how many cars per minute must pass through the other intersections to keep traffic moving? </strong> A) t + 8 cars/min between I _ { 2 } and I _ { 1 } ; t + 3 cars/min between I _ { 1 } and I _ { 3 } B) t + 1 cars/min between I _ { 2 } and I _ { 1 } ; t + 4 cars/min between I _ { 1 } and I _ { 3 } C) t - 2 cars/min between I _ { 2 } and I _ { 1 } ; t + 1 cars/min between I _ { 1 } and I _ { 3 } D) t + 2 cars / \mathrm { min } between \mathrm { I } _ { 2 } and \mathrm { I } _ { 1 } ; \mathrm { t } - 3 cars/min between \mathrm { I } _ { 1 } and \mathrm { I } _ { 3 }

A) t+8t + 8 cars/min between I2I _ { 2 } and I1;t+3I _ { 1 } ; t + 3 cars/min between I1I _ { 1 } and I3I _ { 3 }
B) t+1t + 1 cars/min between I2I _ { 2 } and I1;t+4I _ { 1 } ; t + 4 cars/min between I1I _ { 1 } and I3I _ { 3 }
C) t−2t - 2 cars/min between I2I _ { 2 } and I1;t+1I _ { 1 } ; t + 1 cars/min between I1I _ { 1 } and I3I _ { 3 }
D) t+2t + 2 cars /min/ \mathrm { min } between I2\mathrm { I } _ { 2 } and I1;t−3\mathrm { I } _ { 1 } ; \mathrm { t } - 3 cars/min between I1\mathrm { I } _ { 1 } and I3\mathrm { I } _ { 3 }
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55
Matrix Operations and Their Applications
1 Use Matrix Notation
[−82−105−9−24];a12\left[ \begin{array} { r c c c } - 8 & 2 & - 1 & 0 \\5 & - 9 & - 2 & 4\end{array} \right] ; \mathrm { a } _ { 12 }

A) 2×4;22 \times 4 ; 2
B) 4×2;24 \times 2 ; 2
C) 2×4;52 \times 4 ; 5
D) 4×2;54 \times 2 ; 5
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56
Understand What is Meant by Equal Matrices
[6−8−79]=[xy−7z]\left[ \begin{array} { r r } 6 & - 8 \\- 7 & 9\end{array} \right] = \left[ \begin{array} { r r } x y \\- 7 & z\end{array} \right]

A) x=6;y=−8;z=9x = 6 ; y = - 8 ; z = 9
B) x=6;y=−8;z=−7x = 6 ; y = - 8 ; z = - 7
C) x=6;y=−7;z=9x = 6 ; y = - 7 ; z = 9
D) x=−8;y=6;z=9x = - 8 ; y = 6 ; z = 9
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57
Solve Problems Involving Systems Without Unique Solutions
The nutritional content per ounce for three foods is given in the table below.  Fat (g/oz) Protein (g/oz) Fiber (g/oz) Food A 241 Food B 121 Food C 8165\begin{array}{l|ccc} & \text { Fat }(\mathrm{g} / \mathrm{oz}) & \text { Protein }(\mathrm{g} / \mathrm{oz}) & \text { Fiber }(\mathrm{g} / \mathrm{oz}) \\\hline \text { Food A } & 2 & 4 & 1 \\\text { Food B } & 1 & 2 & 1 \\\text { Food C } & 8 & 16 & 5\end{array}

What combination of these foods can provide exactly 14 grams of fat, 27 grams of protein, and 10 grams of fiber?

A) No possible combination of these foods
B) 3oz3 \mathrm { oz } of Food A; 5oz5 \mathrm { oz } of Food B; 1oz1 \mathrm { oz } of Food CC
C) 7 oz of Food A; 7 oz of Food B; 1 oz of Food CC
D) 4oz4 \mathrm { oz } of Food A;6ozA ; 6 \mathrm { oz } of Food B; 2oz2 \mathrm { oz } of Food CC
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58
Add and Subtract Matrices
Let A=[5−45−5−87]A = \left[ \begin{array} { r r } 5 & - 4 \\ 5 & - 5 \\ - 8 & 7 \end{array} \right] and B=[−76−894−4]B = \left[ \begin{array} { r r } - 7 & 6 \\ - 8 & 9 \\ 4 & - 4 \end{array} \right] . Find A+BA + B

A)
[−22−34−43]\left[ \begin{array} { l l l } - 2 & 2 \\ - 3 & 4 \\ - 4 & 3 \end{array} \right]
B)
[12−1013−14−122]\left[ \begin{array} { r r } 12 & - 10 \\ 13 & - 14 \\ - 12 & 2 \end{array} \right]
C)
[−223−5−4−3]\left[ \begin{array} { r r } - 2 & 2 \\ 3 & - 5 \\ - 4 & - 3 \end{array} \right]
D)
[−2−5−34−43]\left[ \begin{array} { r r } - 2 & - 5 \\ - 3 & 4 \\ - 4 & 3 \end{array} \right]
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59
Add and Subtract Matrices
Let A=[3−5−4]A = \left[ \begin{array} { r } 3 \\ - 5 \\ - 4 \end{array} \right] and B=[−539]B = \left[ \begin{array} { r } - 5 \\ 3 \\ 9 \end{array} \right] . Find A+BA + B .

A)
[−2−25]\left[ \begin{array} { r } - 2 \\ - 2 \\ 5 \end{array} \right]
B)
[−2−25]\left[ \begin{array} { l l l } - 2 & - 2 & 5 \end{array} \right]
C)
[3−5−53−49]\left[ \begin{array} { r r } 3 & - 5 \\ - 5 & 3 \\ - 4 & 9 \end{array} \right]
D)
[256]\left[ \begin{array} { l } 2 \\ 5 \\ 6 \end{array} \right]
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60
Solve Problems Involving Systems Without Unique Solutions
A company that manufactures products A, B, and C does both assembly and testing. The hours needed to assemble and test each product are shown in the table below.  Hours needed  weekly to assemble  Hours needed  weekly to test  Product A 14 Product B 15 Product C 210\begin{array}{l|cc} & \begin{array}{l}\text { Hours needed } \\\text { weekly to assemble }\end{array} & \begin{array}{c}\text { Hours needed } \\\text { weekly to test }\end{array} \\\hline \text { Product A } & 1 & 4 \\\text { Product B } & 1 & 5 \\\text { Product C } & 2 & 10\end{array}

The company has exactly 27 hours per week available for assembly and 120 hours per week available for testing. If the company must produce tt units of Product CC this week, how many units of Products AA and BB can they produce?

A) 15 of Product A; −2t+12- 2 t + 12 of Product B
B) 15 t of Product A; 2t+122 t + 12 of Product BB
C) t+15t + 15 of Product A;t+12A ; t + 12 of Product BB
D) 15 of Product A; 12 of Product B
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61
Solve Matrix Equations
Let A=[9449−36]\mathrm { A } = \left[ \begin{array} { r r } 9 & 4 \\ 4 & 9 \\ - 3 & 6 \end{array} \right] and B=[−2−19−9−54];X−B=A\mathrm { B } = \left[ \begin{array} { r r } - 2 & - 1 \\ 9 & - 9 \\ - 5 & 4 \end{array} \right] ; \quad \mathrm { X } - \mathrm { B } = \mathrm { A }

A)
[73130−810]\left[ \begin{array} { c c } 7 & 3 \\ 13 & 0 \\ - 8 & 10 \end{array} \right]
B)
[115−518214]\left[ \begin{array} { r r } 11 & 5 \\ - 5 & 18 \\ 2 & 14 \end{array} \right]
C)
[73−139−8−10]\left[ \begin{array} { r r } 7 & 3 \\ - 13 & 9 \\ - 8 & - 10 \end{array} \right]

D)
[79130−810]\left[ \begin{array} { r r } 7 & 9 \\ 13 & 0 \\ - 8 & 10 \end{array} \right]
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62
Multiply Matrices
A=[3−2104−3],B=[30−21]A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 3 \end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \end{array} \right]

A) AB\mathrm { AB } is not defined.
B)
[9−63−68−5]\left[\begin{array}{rrr}9 & -6 & 3 \\-6 & 8 & -5\end{array}\right]

C) [9−6−683−5]\left[\begin{array}{rr}9 & -6 \\-6 & 8 \\3 & -5\end{array}\right]
D)
[9004]\left[ \begin{array} { l l } 9 & 0 \\ 0 & 4 \end{array} \right]
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63
Solve Matrix Equations
Let A=[45039−9]\mathrm { A } = \left[ \begin{array} { c r } 4 & 5 \\ 0 & 3 \\ 9 & - 9 \end{array} \right] and B=[9−9−4508];B−X=3 A\mathrm { B } = \left[ \begin{array} { r r } 9 & - 9 \\ - 4 & 5 \\ 0 & 8 \end{array} \right] ; \quad \mathrm { B } - \mathrm { X } = 3 \mathrm {~A}

A)
X=[−3−24−4−4−2735]X = \left[ \begin{array} { c c } - 3 & - 24 \\ - 4 & - 4 \\ - 27 & 35 \end{array} \right]
B)
X=[21641427−19]X = \left[ \begin{array} { r r } 21 & 6 \\ 4 & 14 \\ 27 & - 19 \end{array} \right]
C)
X=[−3−244−4−2735]X = \left[ \begin{array} { r r } - 3 & - 24 \\ 4 & - 4 \\ - 27 & 35 \end{array} \right]
D)
X=[2164149−19]X = \left[ \begin{array} { r r } 21 & 6 \\ 4 & 14 \\ 9 & - 19 \end{array} \right]
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64
Multiply Matrices
A=[1−61−34−2],B=[−1−4−7]A = \left[ \begin{array} { r r r } 1 & - 6 & 1 \\- 3 & 4 & - 2\end{array} \right] , B = \left[ \begin{array} { l } - 1 \\- 4 \\- 7\end{array} \right]

A) [161]\left[ \begin{array} { r } 16 \\ 1 \end{array} \right]
B) AB\mathrm { AB } is not defined.
C) [ 161]\left. \begin{array} { l l } 16 & 1 \end{array} \right]
D)
[1−61−34−2−1−4−7]\left[ \begin{array} { r c r } 1 & - 6 & 1 \\- 3 & 4 & - 2 \\- 1 & - 4 & - 7\end{array} \right]
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65
Multiply Matrices
A=[−1316],B=[0−251−32]A = \left[ \begin{array} { r r } - 1 & 3 \\ 1 & 6 \end{array} \right] , B = \left[ \begin{array} { l l l } 0 & - 2 & 5 \\ 1 & - 3 & 2 \end{array} \right]


A)
[3−716−2017]\left[ \begin{array} { r r r } 3 & - 7 & 1 \\6 & - 20 & 17\end{array} \right]
B) AB\mathrm { AB } is not defined.
C)
[36−7−20117]\left[ \begin{array} { r r } 3 & 6 \\ - 7 & - 20 \\ 1 & 17 \end{array} \right]
D)
[0−6151−1812]\left[ \begin{array} { c c c } 0 & - 6 & 15 \\ 1 & - 18 & 12 \end{array} \right]
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66
Perform Scalar Multiplication
Let A=[1−32]\mathrm { A } = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right] and B=[−13−2]\mathrm { B } = \left[ \begin{array} { r } - 1 \\ 3 \\ - 2 \end{array} \right] . Find A−4 B\mathrm { A } - 4 \mathrm {~B} .

A)
[5−1510]\left[ \begin{array} { r } 5 \\ - 15 \\ 10 \end{array} \right]
B)
[−39−6]\left[ \begin{array} { r } - 3 \\ 9 \\ - 6 \end{array} \right]
C)
[−515−10]\left[ \begin{array} { r } - 5 \\ 15 \\ - 10 \end{array} \right]
D)
[5−64]\left[ \begin{array} { r } 5 \\ - 6 \\ 4 \end{array} \right]
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67
Solve Matrix Equations
Let A=[5−2−12]A = \left[ \begin{array} { r r } 5 & - 2 \\ - 1 & 2 \end{array} \right] and B=[−1−555];X+A=BB = \left[ \begin{array} { r r } - 1 & - 5 \\ 5 & 5 \end{array} \right] ; \quad X + A = B

A)
X=[−6−363]X = \left[ \begin{array} { r r } - 6 & - 3 \\ 6 & 3 \end{array} \right]
B)

X=[−3−636]X = \left[ \begin{array} { r r } - 3 & - 6 \\ 3 & 6 \end{array} \right]
C)
X=[63−6−3]X = \left[ \begin{array} { r r } 6 & 3 \\ - 6 & - 3 \end{array} \right]
D)
X=[3−6−36]X = \left[ \begin{array} { r r } 3 & - 6 \\ - 3 & 6 \end{array} \right]
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68
Multiply Matrices
A=[−65−3−7−8−5−22−7],B=[1−8−15−62−211]A = \left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]

A)
[251513−3799−1422−3−1]\left[ \begin{array} { r r r } 25 & 15 & 13 \\- 37 & 99 & - 14 \\22 & - 3 & - 1\end{array} \right]
B)
[25−37221599−313−14−1]\left[\begin{array}{rrr}25 & -37 & 22 \\15 & 99 & -3 \\13 & -14 & -1\end{array}\right]
C)

[−65−3−7−8−5−22−71−8−15−62−211]\left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7 \\1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]
D)
[−6−403−3548−1042−7]\left[\begin{array}{rrr}-6 & -40 & 3 \\-35 & 48 & -10 \\4 & 2 & -7\end{array}\right]

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69
Solve Matrix Equations
Let A=[7−7−4010−1]\mathrm { A } = \left[ \begin{array} { r r } 7 & - 7 \\ - 4 & 0 \\ 10 & - 1 \end{array} \right] and B=[1000−27−1];4X+A=B\mathrm { B } = \left[ \begin{array} { r r } 10 & 0 \\ 0 & - 2 \\ 7 & - 1 \end{array} \right] ; \quad 4 X + A = B

A)
X=[34741−12−340]X=[−34−74−112340]\mathrm { X } = \left[ \begin{array} { c c } \frac { 3 } { 4 } & \frac { 7 } { 4 } \\ 1 & - \frac { 1 } { 2 } \\ - \frac { 3 } { 4 } & 0 \end{array} \right] \quad \mathrm { X } = \left[ \begin{array} { c c } - \frac { 3 } { 4 } & - \frac { 7 } { 4 } \\ - 1 & \frac { 1 } { 2 } \\ \frac { 3 } { 4 } & 0 \end{array} \right]
B)
X=[−34−74−112340]X=\left[\begin{array}{cc}-\frac{3}{4} & -\frac{7}{4} \\-1 & \frac{1}{2} \\\frac{3}{4} & 0\end{array}\right]

C)
X=[374−2−30]X=\left[\begin{array}{rr}3 & 7 \\4 & -2 \\-3 & 0\end{array}\right]

D)
X=[−37−4230]X = \left[ \begin{array} { r r } - 3 & 7 \\ - 4 & 2 \\ 3 & 0 \end{array} \right]
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70
Model Applied Situations with Matrix Operations
Adjust the contrast by changing the black to dark grey and the light grey to white. Use matrix addition to accomplish this. A) [131131333]+[−1−1−1−1−1−1−1−1−1]=[020020222]\left[ \begin{array} { l l l } 1 & 3 & 1 \\ 1 & 3 & 1 \\ 3 & 3 & 3 \end{array} \right] + \left[ \begin{array} { l l l } - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \\ - 1 & - 1 & - 1 \end{array} \right] = \left[ \begin{array} { l l l } 0 & 2 & 0 \\ 0 & 2 & 0 \\ 2 & 2 & 2 \end{array} \right]
B)
[131131333]+[0−100−10−1−1−1]=[121121222]\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { r r r } 0 & - 1 & 0 \\0 & - 1 & 0 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 1 & 2 & 1 \\1 & 2 & 1 \\2 & 2 & 2\end{array} \right]
C)
[131131333]+[111111111]=[242242444]\left[ \begin{array} { l l l } 1 & 3 & 1 \\ 1 & 3 & 1 \\ 3 & 3 & 3 \end{array} \right] + \left[ \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right] = \left[ \begin{array} { l l l } 2 & 4 & 2 \\ 2 & 4 & 2 \\ 4 & 4 & 4 \end{array} \right]
D)
[131131333]+[0−100−10−1−1−1]=[020020222]\left[ \begin{array} { l l l } 1 & 3 & 1 \\1 & 3 & 1 \\3 & 3 & 3\end{array} \right] + \left[ \begin{array} { r r r } 0 & - 1 & 0 \\0 & - 1 & 0 \\- 1 & - 1 & - 1\end{array} \right] = \left[ \begin{array} { l l l } 0 & 2 & 0 \\0 & 2 & 0 \\2 & 2 & 2\end{array} \right]
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71
Multiply Matrices
A=[−1322],B=[−20−15]A = \left[ \begin{array} { r r } - 1 & 3 \\2 & 2\end{array} \right] , B = \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 5\end{array} \right]

A)
[−115−610]\left[ \begin{array} { l l } - 1 & 15 \\ - 6 & 10 \end{array} \right]
B)
[20−210]\left[ \begin{array} { r r } 2 & 0 \\ - 2 & 10 \end{array} \right]
C)
[2−6−17]\left[ \begin{array} { r r } 2 & - 6 \\ - 1 & 7 \end{array} \right]
D)
[15−110−6]\left[ \begin{array} { l l } 15 & - 1 \\ 10 & - 6 \end{array} \right]
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72
Multiply Matrices

A)
[−68−79][ - 68 - 79 ]
B) AB\mathrm { AB } is not defined.
C)
[−68−79]\left[ \begin{array} { l } - 68 \\ - 79 \end{array} \right]
D)

[−5−8−68−3−6−239]\left[ \begin{array} { r r r } - 5 & - 8 & - 6 \\ 8 & - 3 & - 6 \\ - 2 & 3 & 9 \end{array} \right]
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73
Multiply Matrices
A=[−1−95],B=[1−59562−5−9−8]A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]

A) [−71−94−67][ - 71 - 94 - 67 ]
B)
[−71−94−67]\left[ \begin{array} { l } - 71 \\ - 94 \\ - 67 \end{array} \right]
C)
[−1−951−59562−5−9−8]\left[ \begin{array} { r r r } - 1 & - 9 & 5 \\ 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]
D)
[−14545−5−5410581−40]\left[ \begin{array} { r r r } - 1 & 45 & 45 \\ - 5 & - 54 & 10 \\ 5 & 81 & - 40 \end{array} \right]
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74
Perform Scalar Multiplication
Let A=[−32]A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right] and B=[10]B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right] . Find 2A+3B2 A + 3 B .

A) [−34][ - 34 ]
B) [−64][ - 64 ]
C) [−54]\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]
D) [02]\left[ \begin{array} { l l } 0 & 2 \end{array} \right]
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75
Multiply Matrices
A=[−628],B=[70−3]A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]

A) [−66][ - 66 ]
B) [330][ 330 ]
C) [−420−24][ - 420 - 24 ]
D)
[−420−24]\left[ \begin{array} { r } - 42 \\0 \\- 24\end{array} \right]
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76
Multiply Matrices
A=[3−2104−2],B=[50−22]A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 2 \end{array} \right] , B = \left[ \begin{array} { r r } 5 & 0 \\ - 2 & 2 \end{array} \right]

A) AB\mathrm { AB } is not defined.
B) [15−105−612−6]\left[ \begin{array} { r r r } 15 & - 10 & 5 \\ - 6 & 12 & - 6 \end{array} \right]
C)
[15−6−10125−6]\left[ \begin{array} { r r } 15 & - 6 \\- 10 & 12 \\5 & - 6\end{array} \right]
D)
[15008]\left[\begin{array}{rr}15 & 0 \\0 & 8\end{array}\right]


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77
Solve Matrix Equations
Let A=[1−1−3−1]\mathrm { A } = \left[ \begin{array} { r } 1 - 1 \\ - 3 - 1 \end{array} \right] and B=[−3−41−5];X+A=B\mathrm { B } = \left[ \begin{array} { r } - 3 - 4 \\ 1 - 5 \end{array} \right] ; \quad \mathrm { X } + \mathrm { A } = \mathrm { B }

A)
X=[−4−34−4]X = \left[ \begin{array} { r r } - 4 & - 3 \\ 4 & - 4 \end{array} \right]
B)
X=[−3−4−44]X = \left[ \begin{array} { r r } - 3 & - 4 \\ - 4 & 4 \end{array} \right]
C)
X=[4−4−4−3]X = \left[ \begin{array} { r r } 4 & - 4 \\- 4 & - 3 \end{array} \right]
D)
X=[−44−3−4]X = \left[ \begin{array} { r r } - 4 & 4 \\- 3 & - 4\end{array} \right]
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78
Solve Matrix Equations
Let A=[21−23001−22]\mathrm { A } = \left[ \begin{array} { r r r } 2 & 1 & - 2 \\ 3 & 0 & 0 \\ 1 & - 2 & 2 \end{array} \right] and B=[−1−2−2011302];2 B−2 A=X\mathrm { B } = \left[ \begin{array} { r r r } - 1 & - 2 & - 2 \\ 0 & 1 & 1 \\ 3 & 0 & 2 \end{array} \right] ; \quad 2 \mathrm {~B} - 2 \mathrm {~A} = \mathrm { X }

A)
X=[−6−60−622440]X = \left[ \begin{array} { r r r } - 6 & - 6 & 0 \\ - 6 & 2 & 2 \\ 4 & 4 & 0 \end{array} \right]
B)
X=[−622440−6−60]X = \left[ \begin{array} { r r r } - 6 & 2 & 2 \\ 4 & 4 & 0 \\ - 6 & - 6 & 0 \end{array} \right]
C)
X=[−6−60−611440]X = \left[ \begin{array} { r r r } - 6 & - 6 & 0 \\ - 6 & 1 & 1 \\ 4 & 4 & 0 \end{array} \right]
D)
X=[−611440−6−60]X = \left[ \begin{array} { r r r } - 6 & 1 & 1 \\ 4 & 4 & 0 \\ - 6 & - 6 & 0 \end{array} \right]
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79
Multiply Matrices
A=[13−1203],B=[30−1103]A = \left[ \begin{array} { r r r } 1 & 3 & - 1 \\2 & 0 & 3\end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\- 1 & 1 \\0 & 3\end{array} \right]

A)
[0069]\left[ \begin{array} { l l } 0 & 0 \\ 6 & 9 \end{array} \right]
B) ABA B is not defined.
C)
[3−30009]\left[ \begin{array} { r r r } 3 & - 3 & 0 \\ 0 & 0 & 9 \end{array} \right]
D)
[0096]\left[ \begin{array} { l l } 0 & 0 \\ 9 & 6 \end{array} \right]
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80
Perform Scalar Multiplication
Let A=[−9−38−1−68966]A = \left[ \begin{array} { r r r } - 9 & - 3 & 8 \\ - 1 & - 6 & 8 \\ 9 & 6 & 6 \end{array} \right] and B=[−389−5−41619]B = \left[ \begin{array} { r r r } - 3 & 8 & 9 \\ - 5 & - 4 & 1 \\ 6 & 1 & 9 \end{array} \right] . Find 2A - 4B.

A)
[−6−38−2018412−68−24]\left[ \begin{array} { r r r } - 6 & - 38 & - 20 \\ 18 & 4 & 12 \\ - 6 & 8 & - 24 \end{array} \right]
B)
[−21225−7−1617241321]\left[\begin{array}{rrr}-21 & 2 & 25 \\-7 & -16 & 17 \\24 & 13 & 21\end{array}\right]
C)
[−12517−6−10915715]\left[ \begin{array} { r r r } - 12 & 5 & 17 \\ - 6 & - 10 & 9 \\ 15 & 7 & 15 \end{array} \right]
D)
[−12−6155−10717915]\left[ \begin{array} { r r r } - 12 & - 6 & 15 \\ 5 & - 10 & 7 \\ 17 & 9 & 15 \end{array} \right]
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