Question 1

Find $A ^ { 2 }$ if possible.
-$A = \left[ \begin{array} { r r r } 
9 & - 1 & 8 \
8 & - 5 & - 2 \
- 1 & 5 & 1
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 65 & 34 & 30 \ 36 & 7 & - 19 \ 82 & 72 & - 17 \end{array} \right] \quad$ 
B)  Not possible $\quad$ 
C)  $\left[ \begin{array} { r r r } 65 & 36 & 82 \ 9 & 7 & 72 \ 30 & - 19 & - 17 \end{array} \right] \quad$

D)  $\left[ \begin{array} { r r r } 81 & 1 & 64 \ 64 & 25 & 4 \ 1 & 25 & 1 \end{array} \right]$

Question 2

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Associated with every square matrix A is a real number denoted by $$| A |$$ called the________ of A.

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Determine the inverse of the given matrix, if possible. Otherwise, state the matrix is singular.
-$A = \left[ \begin{array} { c c c } 
- \frac { 1 } { 4 } & - \frac { 1 } { 2 } & - \frac { 1 } { 4 } \
- \frac { 3 } { 8 } & - \frac { 1 } { 4 } & \frac { 1 } { 8 } \
\frac { 17 } { 16 } & \frac { 7 } { 8 } & \frac { 5 } { 16 }
\end{array} \right]$

Accepted Answer

A)  $A ^ { - 1 } = \left[ \begin{array} { l l l } 3 & - 1 & 2 \ 4 & - 3 & 2 \ 1 & - 5 & 2 \end{array} \right]$ 
B)  $A ^ { - 1 } = \left[ \begin{array} { l l l } 3 & - 4 & 1 \ 1 & - 3 & 5 \ 2 & - 2 & 2 \end{array} \right]$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 3 & 1 & 2 \ - 4 & - 3 & - 2 \ 1 & 5 & 2 \end{array} \right]$ 
D)  Singular matrix

Question 4

Find AB, if possible.
-$A = \left[ \begin{array} { r r r } - 7 & 7 & - 5 \ 3 & 9 & 3 \ 8 & - 9 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { r } 5 \ 8 \ - 1 \end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { r } 7 \ 27 \ 12 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } 26 & 84 & - 33 \end{array} \right]$ 
C)  Not possible 
D)  $\left[ \begin{array} { r } 26 \ 84 \ - 33 \end{array} \right]$

Question 5

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$\begin{array} { l } 
3 x + 2 y + 7 z = 38 \
9 x + 6 y + 21 z = 114 \
0.6 x + 0.4 y + 1.4 z = 7.6
\end{array}$

Accepted Answer

A) { } 
B)  { ( x , y , z )|  3 x +2 y +7 z = 38 } 
C)  $\{ ( -1,-4,7 ) \}$ 
D)  $\{ ( 1,4,-7) \}$

Question 6

Find A + B.
-$A = \left[ \begin{array} { r r r } 
- 6 & - 9 & - 5 \
2 & - 1 & - 2
\end{array} \right] , B = \left[ \begin{array} { r r } 
- 9 & - 6 \
- 5 & - 3 \
- 5 & 6
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { r r } - 12 & - 7 \ - 12 & - 6 \ 1 & - 7 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 15 & - 15 \ - 3 & - 4 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r r } - 15 & - 15 & - 5 \ - 3 & - 4 & - 2 \end{array} \right]$ 
D)  Undefined

Question 7

Choose the one alternative that best completes the statement or answers the question.
For the given augmented matrix, determine the number of solutions to the corresponding system of
equations.
-$$\left[ \begin{array} { r r r | r } 
1 & 0 & 7 & - 2 \
0 & 1 & - 8 & 1 \
0 & 0 & 0 & 0
\end{array} \right]$$

Accepted Answer

A)  One solution 
B)  Infinitely many solutions 
C)  No solution

Question 8

Find C - A + B
-$A = \left[ \begin{array} { r r } 5 & - 3 \ 6 & 6 \ 6 & \sqrt { 14 } \end{array} \right] , B = \left[ \begin{array} { r r } - 4 & 3 \ 6.5 & 1 \ \frac { 1 } { 3 } & \sqrt { 11 } \end{array} \right] , C = \left[ \begin{array} { l l } 3 & 2 \ 1 & - \frac { 1 } { 3 } \ 9 & 2 \end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { c c } - 3 & 10 \ 3.5 & - 6 \ - \frac { 8 } { 3 } & 2 + \sqrt { 2 } - \sqrt { 11 } \end{array} \right]$ 
B)  $\left[ \begin{array} { c c } - 6 & 8 \ 1.5 & - \frac { 16 } { 3 } \ \frac { 10 } { 3 } & 2 - \sqrt { 14 } + \sqrt { 11 } \end{array} \right]$ 
C)  $\left[ \begin{array} { c c } 6 & 14 \ 2 + \sqrt { 11 } & 9.6 \ 27 & - 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { c r } 17 & 3 \ 2 + \sqrt { 5 } & 4.2 \ 9 & \frac { 10 } { 3 } \end{array} \right]$

Question 9

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
- 5 y = 24 - 3 x \
2 ( x - 4 y ) = 49 - y
\end{array}$$

Accepted Answer

A)  $\{ ( 7,9 ) \}$ 
B)  $\{ ( - 7 , - 9 ) \}$ 
C)  $\{ ( 9,7 ) \}$ 
D)  $\{ ( - 9 , - 7 ) \}$

Question 10

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-True or false? A system of linear equations in three variables may have exactly one solution.

Accepted Answer

A) True 
 B)False

Question 11

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-An accountant checks the reported earnings for a concert venue for three different performers against the number of tickets sold. $\begin{array}{ccccc}
\text { Performer } & \begin{array}{c}
\text { Children } \
\text { Tickets }
\end{array} & \begin{array}{c}
\text { Student } \
\text { Tickets }
\end{array} & \begin{array}{c}
\text { General } \
\text { Admission }
\end{array} & \begin{array}{c}
\text { Total } \
\text { Revenue }
\end{array} \
\mathbf{1} & 1,000 & 250 & 1,000 & \$ 24,750 \
\mathbf{2} & 400 & 300 & 300 & \$ 10,700 \
\mathbf{3} & 400 & 300 & 300 & \$ 8,700
\end{array}$
 Let $x , y$, and $z$ represent the cost for children tickets, student tickets, and general admission tickets, respectively. Set up an augmented matrix for the system and solve for $x , y$, and $z$. Explain what the accountant knows about the reported earnings.

Accepted Answer

A) {  } ; The system of equations reduces to a contradiction. There are no values for $x , y$, and $z$ that can simultaneously meet the conditions of this problem. 
B)  $\{ ( 7,7,16 ) \}$; There were $\$ 7$ children's tickets, $\$ 7$ student tickets, and $\$ 16$ general admission tickets sold. 
C)  $\{ ( 8,8,17 ) \}$; There were $\$ 8$ children's tickets, $\$ 8$ student tickets, and $\$ 17$ general admission tickets sold. 
D)  $\{ ( 6,6,15 ) \}$; There were $\$ 6$ children's tickets, $\$ 6$ student tickets, and $\$ 15$ general admission tickets sold.

Question 12

Solve the problem.
-A math course has 4 exams weighted 15%, 10%, 20%, and 55%, respectively, toward the final grade. Suppose that a student earns grades of 75, 84, 92, and 86, respectively, on the four exams.
The weights are given in matrix W and the test grades are given in matrix G. Compute WG and Interpret the result. 
$W=\left[\begin{array}{cccc}
0.15 & 0.10 & 0.20 & 0.55
\end{array}\right], \quad G=\left[\begin{array}{c}
75 \
84 \
92 \
86
\end{array}\right]
 \begin{array}{c}
\text { Test 1 } \
\text { Test 2 } \
\text { Test 3 } \
\text { Test 4 }
\end{array}$

Accepted Answer

A)  $[ 50.55 ]$; The value $50.55$ is the student's overall course grade. 
B)  $[ 85.35 ]$; The value $85.35$ is the student's overall course grade. 
C)  $[ 80.15 ]$; The value $80.15$ is the student's overall course grade. 
D)  $[ 84.25 ]$; The value $84.25$ is the student's overall course grade.

Question 13

Choose the one alternative that best completes the statement or answers the question.
Evaluate the determinant of the given matrix.
-$$A = \left[ \begin{array} { r r } 
\log m & \log m \
2 & - 9
\end{array} \right]$$

Accepted Answer

A)  $11 \log m$ 
B)  $- 7 \log m$ 
C)  $- 11 \log m$ 
D)  $- 18 \log m$

Question 14

Perform the elementary row operation on the given matrix.
-$3 R _ { 2 } + R _ { 1 } \rightarrow R _ { 1 }$
$\left[ \begin{array} { r r | r } 
4 & 5 & 1 \
- 2 & - 7 & - 1
\end{array} \right]$

Accepted Answer

A)  $ \left[\begin{array}{rr|r}2 & -2 & 0 \ -2 & -7 & -1\end{array}\right] $ 
B)  $ \left[\begin{array}{rr|r}-10 & -26 & -4 \ -2 & -7 & -1\end{array}\right] $ 
C)  $ \left[\begin{array}{rr|r}-2 & -16 & -2 \ -2 & -7 & -1\end{array}\right] $ 
D)  $ \left[\begin{array}{rr|r}4 & 5 & 1 \ 10 & 8 & 2\end{array}\right] $

Question 15

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-A matrix with the same number of rows and columns is called a ________matrix.

Accepted Answer

The answer of Write the word or phrase that best...

Question 16

Write a system of linear equations represented by the augmented matrix.
-$$\left[ \begin{array} { r r r | r } 
1 & 0 & 0 & 7 \
0 & 1 & 0 & - 9 \
0 & 0 & 1 & \frac { 1 } { 9 }
\end{array} \right]$$

Accepted Answer

A)  $x = - 7$
$y = 9$
$z = - \frac { 1 } { 9 }$ 
B)  $x = - 7$
$y = - 9$
$z = \frac { 1 } { 9 }$ 
C)  $x = 7$
$y = 9$
$z = \frac { 1 } { 9 }$ 
D)  $x = 7$
$y = - 9$
$z = \frac { 1 } { 9 }$

Question 17

Determine the inverse of the given matrix, if possible. Otherwise, state the matrix is singular.
-$A=\left[\begin{array}{rrrr}
-5 & 5 & 3 & -4 \
0 & 1 & -2 & 0 \
3 & 2 & 1 & 3 \
0 & 1 & 1 & 0
\end{array}\right]$

Accepted Answer

A) $A^{-1}=\left[\begin{array}{rrrr}
-1 & -\frac{10}{9} & -\frac{4}{3} & -\frac{59}{9} \
0 & -\frac{1}{3} & 0 & -\frac{2}{3} \
0 & \frac{1}{3} & 0 & -\frac{1}{3} \
1 & \frac{11}{9} & \frac{5}{3} & \frac{64}{9}
\end{array}\right]$ 
B) $A^{-1}=\left[\begin{array}{rrrr}
-1 & \frac{10}{9} & -\frac{4}{3} & \frac{59}{9} \
0 & \frac{1}{3} & 0 & \frac{2}{3} \
0 & -\frac{1}{3} & 0 & \frac{1}{3} \
1 & -\frac{11}{9} & \frac{5}{3} & -\frac{64}{9}
\end{array}\right]$ 
C) $\text { Singular matrix }$ 
D) $A^{-1}=\left[\begin{array}{cccc}
1 & \frac{10}{9} & \frac{4}{3} & \frac{59}{9} \
0 & \frac{1}{3} & 0 & \frac{2}{3} \
0 & -\frac{1}{3} & 0 & \frac{1}{3} \
-1 & -\frac{11}{9} & -\frac{5}{3} & -\frac{64}{9}
\end{array}\right]$

Question 18

The equation $$\left| \begin{array} { l l l } 
x & y & 1 \
x _ { 1 } & y _ { 1 } & 1 \
x _ { 2 } & y _ { 2 } & 1
\end{array} \right| = 0$$ = 0 represents an equation of the line passing through distinct points $\left( x _ { 1 } , y _ { 1 } \right)$ and $\left( x _ { 2 } , y _ { 2 } \right)$ a. Use the determinant equation to write an equation of the line passing through the given points.
b. Write the equation of the line in slope-intercept form.
-$( - 2,6 )$ and $( - 3,2 )$

Accepted Answer

A)  a. $\left| \begin{array} { r r r } x & y & 1 \ - 2 & 6 & 1 \ - 3 & 2 & 1 \end{array} \right| = 0 ;$ b. $y = - 8 x - 10$ 
B)  $\mathbf { a } . \left| \begin{array} { c c c } x & y & 1 \ - 2 & - 3 & 1 \ 6 & 2 & 1 \end{array} \right| = 0 ;$ b. $y = 4 x + 14$ 
C)  $\mathbf { a } . \left| \begin{array} { c c c } x & y & 1 \ - 2 & - 3 & 1 \ 6 & 2 & 1 \end{array} \right| = 0 ;$ b. $y = - 8 x - 10$ 
D)  a. $\left| \begin{array} { r r r } x & y & 1 \ - 2 & 6 & 1 \ - 3 & 2 & 1 \end{array} \right| = 0 ;$ b. $y = 4 x + 14$

Question 19

Identify the elements on the main diagonal. $$\left[ \begin{array} { r r r | r } 
- 4 & - 1 & 1 & 8 \
2 & 0 & 5 & 11 \
0 & 1 & - 7 & - 6
\end{array} \right]$$

Accepted Answer

The answer of Identify the elements on the main diagonal....

Question 20

Solve the system using Gaussian elimination or Gauss-Jordan elimination.
-$$\begin{array} { l } 
w - 2 x + 3 y = 8 \
- 5 x - 5 y = - 25 \
- 2 x + 3 y - 4 z = 17 \
- 3 w - 5 x + 2 y - 3 z = - 4
\end{array}$$

Accepted Answer

A)  $\{ ( - 3 , - 2,3 , - 3 ) \}$ 
B)  $\{ ( - 3 , - 2 , - 3,3 ) \}$ 
C)  $\{ ( 3,2,3 , - 3 ) \}$ 
D)  $\{ ( 3,2 , - 3,3 ) \}$

Exam 6: Matrices and Determinants and Applications

Find $A ^ { 2 }$ if possible. - $A = \left[ \begin{array} { r r r } 9 & - 1 & 8 \\8 & - 5 & - 2 \\- 1 & 5 & 1\end{array} \right]$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Associated with every square matrix A is a real number denoted by $| A |$ called the________ of A.

Find AB, if possible. - $A = \left[ \begin{array} { r r r } - 7 & 7 & - 5 \\ 3 & 9 & 3 \\ 8 & - 9 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { r } 5 \\ 8 \\ - 1 \end{array} \right]$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } 3 x + 2 y + 7 z = 38 \\9 x + 6 y + 21 z = 114 \\0.6 x + 0.4 y + 1.4 z = 7.6\end{array}$

Find A + B. - $A = \left[ \begin{array} { r r r } - 6 & - 9 & - 5 \\2 & - 1 & - 2\end{array} \right] , B = \left[ \begin{array} { r r } - 9 & - 6 \\- 5 & - 3 \\- 5 & 6\end{array} \right]$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } - 5 y = 24 - 3 x \\2 ( x - 4 y ) = 49 - y\end{array}$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -True or false? A system of linear equations in three variables may have exactly one solution.

Choose the one alternative that best completes the statement or answers the question. Evaluate the determinant of the given matrix. - $A = \left[ \begin{array} { r r } \log m & \log m \\2 & - 9\end{array} \right]$

Perform the elementary row operation on the given matrix. - $3 R _ { 2 } + R _ { 1 } \rightarrow R _ { 1 }$ $\left[ \begin{array} { r r | r } 4 & 5 & 1 \\- 2 & - 7 & - 1\end{array} \right]$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A matrix with the same number of rows and columns is called a ________matrix.

Write a system of linear equations represented by the augmented matrix. - $\left[ \begin{array} { r r r | r } 1 & 0 & 0 & 7 \\0 & 1 & 0 & - 9 \\0 & 0 & 1 & \frac { 1 } { 9 }\end{array} \right]$

Determine the inverse of the given matrix, if possible. Otherwise, state the matrix is singular. - $A=\left[\begin{array}{rrrr}-5 & 5 & 3 & -4 \\0 & 1 & -2 & 0 \\3 & 2 & 1 & 3 \\0 & 1 & 1 & 0\end{array}\right]$

Identify the elements on the main diagonal. $\left[ \begin{array} { r r r | r } - 4 & - 1 & 1 & 8 \\2 & 0 & 5 & 11 \\0 & 1 & - 7 & - 6\end{array} \right]$

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - $\begin{array} { l } w - 2 x + 3 y = 8 \\- 5 x - 5 y = - 25 \\- 2 x + 3 y - 4 z = 17 \\- 3 w - 5 x + 2 y - 3 z = - 4\end{array}$

Exam 6: Matrices and Determinants and Applications

Find A2A ^ { 2 }A2 if possible. - A=[9−188−5−2−151]A = \left[ \begin{array} { r r r } 9 & - 1 & 8 \\8 & - 5 & - 2 \\- 1 & 5 & 1\end{array} \right]A=​98−1​−1−55​8−21​​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Associated with every square matrix A is a real number denoted by ∣A∣| A |∣A∣ called the________ of A.

Find AB, if possible. - A=[−77−53938−91]A = \left[ \begin{array} { r r r } - 7 & 7 & - 5 \\ 3 & 9 & 3 \\ 8 & - 9 & 1 \end{array} \right]A=​−738​79−9​−531​​ and B=[58−1]B = \left[ \begin{array} { r } 5 \\ 8 \\ - 1 \end{array} \right]B=​58−1​​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - 3x+2y+7z=389x+6y+21z=1140.6x+0.4y+1.4z=7.6\begin{array} { l } 3 x + 2 y + 7 z = 38 \\9 x + 6 y + 21 z = 114 \\0.6 x + 0.4 y + 1.4 z = 7.6\end{array}3x+2y+7z=389x+6y+21z=1140.6x+0.4y+1.4z=7.6​

Solve the system using Gaussian elimination or Gauss-Jordan elimination. - −5y=24−3x2(x−4y) =49−y\begin{array} { l } - 5 y = 24 - 3 x \\2 ( x - 4 y ) = 49 - y\end{array}−5y=24−3x2(x−4y) =49−y​