Question 1

Solve a System of Linear Equations in Three Variables Using Cramer's Rule
-$$\begin{array} { l } 
3 x + 7 z = 31 \
5 x + 3 y + 2 z = 34 \
- 9 x - 5 y \quad = - 44
\end{array}$$

Accepted Answer

A)  $\{ ( 1,7,4 ) \}$ 
B)  $\{ ( 7,4,7 ) \}$ 
C)  $\{ ( 1 , - 7 , - 4 ) \}$ 
D)  $\{ ( 2,5,4 ) \}$ 
A)  $\{ ( 1,7,4 ) \}$ 
B)  $\{ ( 7,4,7 ) \}$ 
C)  $\{ ( 1 , - 7 , - 4 ) \}$ 
D)  $\{ ( 2,5,4 ) \}$

Question 2

Encode and Decode Messages
-Use the coding matrix $A = \left[ \begin{array} { r r r } 1 & 1 & 1 \ - 1 & 1 & 2 \ 1 & 2 & 3 \end{array} \right]$ and its inverse $A ^ { - 1 } = \left[ \begin{array} { r r r } - 1 & - 1 & 1 \ 5 & 2 & - 3 \ - 3 & - 1 & 2 \end{array} \right]$ to decode the cryptogram $\left[ \begin{array} { l l l } 37 & 16 & 35 \ 38 & 20 & 4 \ 82 & 40 & 60 \end{array} \right]$.

Accepted Answer

A)  GOOD_LUCK 
B)  STAY_CALM 
C)  LOOK_DOWN 
D)  HELP_THEM 
A)  GOOD_LUCK 
B)  STAY_CALM 
C)  LOOK_DOWN 
D)  HELP_THEM

Question 3

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { r r r r } 
2 & 0 & 1 & 1 \
1 & 1 & - 1 & - 1 \
- 1 & - 2 & 1 & 0 \
0 & 0 & 1 & 1
\end{array} \right] , \quad B = \left[ \begin{array} { r r r r } 
2 & 0 & 0 & - 2 \
- 2 & 1 & 0 & 3 \
- 2 & 2 & 1 & 5 \
2 & - 2 & - 1 & - 3
\end{array} \right]$$

Accepted Answer

A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$ Find the inverse of the matrix, if possible. 
A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$ Find the inverse of the matrix, if possible.

Question 4

Understand What is Meant by Equal Matrices
-$$\left[ \begin{array} { r r } 
x + 3 & y + 4 \
7 & 5
\end{array} \right] = \left[ \begin{array} { r r } 
7 & - 3 \
7 & z
\end{array} \right]$$

Accepted Answer

A)  $x = 4 ; y = - 7 ; z = 5$ 
B)  $x = - 4 ; y = 7 ; z = - 5$ 
C)  $x = 7 ; y = - 3 ; z = 5$ 
D)  $x = 4 ; y = 5 ; z = 7$ 
A)  $x = 4 ; y = - 7 ; z = 5$ 
B)  $x = - 4 ; y = 7 ; z = - 5$ 
C)  $x = 7 ; y = - 3 ; z = 5$ 
D)  $x = 4 ; y = 5 ; z = 7$

Question 5

Encode and Decode Messages
-Use the coding matrix $A = \left[ \begin{array} { r r } - 1 & - 3 \ 2 & 5 \end{array} \right]$ to encode the message CARE.

Accepted Answer

A)  $\left[ \begin{array} { r r } - 6 & - 33 \ 11 & 61 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 57 & - 16 \ 96 & 27 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 18 & 105 \ - 7 & - 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { l r } - 1 & - 8 \ - 4 & - 29 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r } - 6 & - 33 \ 11 & 61 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 57 & - 16 \ 96 & 27 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 18 & 105 \ - 7 & - 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { l r } - 1 & - 8 \ - 4 & - 29 \end{array} \right]$

Question 6

Use Determinants to Identify Inconsistent Systems and Systems with Dependent Equations
-$$\begin{array} { l } 
3 x + y = 10 \
6 x + 2 y = 20
\end{array}$$

Accepted Answer

A)  system contains dependent equations 
B)  system is inconsistent 
A)  system contains dependent equations 
B)  system is inconsistent

Question 7

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { l l } 
10 & 1 \
- 1 & 0
\end{array} \right] , \quad B = \left[ \begin{array} { r r } 
0 & 1 \
- 1 & 10
\end{array} \right]$$

Accepted Answer

A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$ 
A)  $B \neq A ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$

Question 8

Use Matrices and Gauss-Jordan Elimination to Solve Systems
-$$\begin{array} { r } 
- 4 x + 7 y - z = 39 \
x + 6 y + 9 z = 62 \
- 5 x + y + z = - 1
\end{array}$$

Accepted Answer

A)  $\{ ( 2,7,2 ) \}$ 
B)  $\{ ( 2,2,7 ) \}$ 
C)  $\{ ( - 2,7,4 ) \}$ 
D)  $\{ ( 4,7 , - 2 ) \}$ 
A)  $\{ ( 2,7,2 ) \}$ 
B)  $\{ ( 2,2,7 ) \}$ 
C)  $\{ ( - 2,7,4 ) \}$ 
D)  $\{ ( 4,7 , - 2 ) \}$

Question 9

Perform Scalar Multiplication
-Let $A = \left[ \begin{array} { r r r } - 9 & - 3 & 8 \ - 1 & - 6 & 8 \ 9 & 6 & 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 3 & 8 & 9 \ - 5 & - 4 & 1 \ 6 & 1 & 9 \end{array} \right]$. Find 2A - 4B.

Accepted Answer

A) 
$\left[ \begin{array} { r r r } - 6 & - 38 & - 20 \ 18 & 4 & 12 \ - 6 & 8 & - 24 \end{array} \right]$ 
B) 
$\left[\begin{array}{rrr}
-21 & 2 & 25 \
-7 & -16 & 17 \
24 & 13 & 21
\end{array}\right]$ 
C) 
$\left[ \begin{array} { r r r } - 12 & 5 & 17 \ - 6 & - 10 & 9 \ 15 & 7 & 15 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } - 12 & - 6 & 15 \ 5 & - 10 & 7 \ 17 & 9 & 15 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r r } - 6 & - 38 & - 20 \ 18 & 4 & 12 \ - 6 & 8 & - 24 \end{array} \right]$ 
B) 
$\left[\begin{array}{rrr}
-21 & 2 & 25 \
-7 & -16 & 17 \
24 & 13 & 21
\end{array}\right]$ 
C) 
$\left[ \begin{array} { r r r } - 12 & 5 & 17 \ - 6 & - 10 & 9 \ 15 & 7 & 15 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } - 12 & - 6 & 15 \ 5 & - 10 & 7 \ 17 & 9 & 15 \end{array} \right]$

Question 10

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { l l l } 
1 & 0 & 0 \
1 & 1 & 0 \
1 & 1 & 1
\end{array} \right] , \quad B = \left[ \begin{array} { r r r } 
1 & 0 & 0 \
- 1 & 1 & 0 \
0 & - 1 & 1
\end{array} \right]$$

Accepted Answer

A)  $B = A ^ { - 1 }$ 
B)  $B \neq A ^ { - 1 }$ 
A)  $B = A ^ { - 1 }$ 
B)  $B \neq A ^ { - 1 }$

Question 11

Evaluate a Third-Order Determinant
-The equation of a line passing through two distinct points $\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right)$ and $\left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right)$ is given by $\left| \begin{array} { l l l } x & y & 1 \ x _ { 2 } & y _ { 2 } & 1 \ x _ { 3 } & y _ { 3 } & 1 \end{array} \right| = 0$. Use the determinant to write an equation for the line passing through $( 9 , - 5 )$ and $( - 8,3 )$. Express the line's equation in standard form.

Accepted Answer

A)  $- 8 x - 17 y - 13 = 0$ 
B)  $- 5 x - 8 y + 27 = 0$ 
C)  $8 x + 17 y - 13 = 0$ 
D)  $3 x + 9 y + 40 = 0$ 
A)  $- 8 x - 17 y - 13 = 0$ 
B)  $- 5 x - 8 y + 27 = 0$ 
C)  $8 x + 17 y - 13 = 0$ 
D)  $3 x + 9 y + 40 = 0$

Question 12

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.

Accepted Answer

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 
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 13

Encode and Decode Messages
-Use the coding matrix $A = \left[ \begin{array} { r r r } 1 & - 2 \ 1 & 2 & 3 \ 1 & 1 & 1 \end{array} \right]$ to encode the message COME_HERE.

Accepted Answer

A)  $\left[ \begin{array} { r r r } - 23 & - 11 & - 5 \ 72 & 29 & 56 \ 31 & 13 & 28 \end{array} \right]$ 
B)   
C)  $\left[ \begin{array} { r r r } 19 & 27 & - 21 \ - 14 & - 30 & 39 \ 8 & 11 & - 13 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } - 7 & - 21 & 3 \ 28 & 69 & 44 \ 13 & 33 & 26 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } - 23 & - 11 & - 5 \ 72 & 29 & 56 \ 31 & 13 & 28 \end{array} \right]$ 
B)   
C)  $\left[ \begin{array} { r r r } 19 & 27 & - 21 \ - 14 & - 30 & 39 \ 8 & 11 & - 13 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } - 7 & - 21 & 3 \ 28 & 69 & 44 \ 13 & 33 & 26 \end{array} \right]$

Question 14

Evaluate a Third-Order Determinant
-$$\left| \begin{array} { l l l } 
7 & 0 & 0 \
5 & 9 & 6 \
8 & 7 & 2
\end{array} \right|$$

Accepted Answer

A)  $- 168$ 
B)  420 
C)  168 
D)  $- 163$ 
A)  $- 168$ 
B)  420 
C)  168 
D)  $- 163$

Question 15

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.
-$\left[ \begin{array} { r r r r | r } 1 & 1 & - 1 & 1 & - 3 \ 0 & 1 & - 4 & 4 & 0 \ 0 & 0 & 1 & 6 & 17 \ 0 & 0 & 0 & 1 & 7 \end{array} \right]$

Accepted Answer

A)  $\{ ( 93 , - 128 , - 25,7 ) \}$ 
B)  $\{ ( - 3,0,17,7 ) \}$ 
C)  $\{ ( - 5 , - 1,10,6 ) \}$ 
D)  $\{ ( 7 , - 25 , - 128,93 ) \}$ 
A)  $\{ ( 93 , - 128 , - 25,7 ) \}$ 
B)  $\{ ( - 3,0,17,7 ) \}$ 
C)  $\{ ( - 5 , - 1,10,6 ) \}$ 
D)  $\{ ( 7 , - 25 , - 128,93 ) \}$

Question 16

Matrix Operations and Their Applications
1 Use Matrix Notation
-$$\left[ \begin{array} { c c c c c } 
3 & 7 & 6 & 7 & - 3 \
3 & - e & - 15 & - 13 & \pi \
- 6 & 9 & - 12 & 12 & 8 \
\frac { 1 } { 3 } & 9 & 6 & - 1 & 5
\end{array} \right] ; a _ { 34 }$$

Accepted Answer

A)  $4 \times 5 ; 12$ 
B)  $5 \times 4 ; 6$ 
C)  $20 ; 8$ 
D)  $4 \times 4 ; - 12$ 
A)  $4 \times 5 ; 12$ 
B)  $5 \times 4 ; 6$ 
C)  $20 ; 8$ 
D)  $4 \times 4 ; - 12$

Question 17

Add and Subtract Matrices
-A)
$$\left[ \begin{array} { r r } 
- 8 & 1 \
- 17 & 0 \
5 & - 6
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { r r } -8 & 1 \ -17 & 8\ 5 & -6 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 1 & 4 \ 7 & 8 \ 11 & 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 1 & 1 \ 7 & 0 \ 5 & - 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 3 & - 2 \ 7 & 0 \ - 5 & 6 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } -8 & 1 \ -17 & 8\ 5 & -6 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 1 & 4 \ 7 & 8 \ 11 & 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 1 & 1 \ 7 & 0 \ 5 & - 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 3 & - 2 \ 7 & 0 \ - 5 & 6 \end{array} \right]$

Question 18

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$A = \left[ \begin{array} { l l } - 4 & 6 \ - 5 & 0 \end{array} \right]$

Accepted Answer

A) $\left[ \begin{array} { c } 0 - \frac { 1 } { 5 } \ \frac { 1 } { 6 } - \frac { 2 } { 15 } \end{array} \right]$ 
B) $\left[ \begin{array} { c r } 0 & \frac { 1 } { 5 } \ - \frac { 1 } { 6 } & - \frac { 2 } { 15 } \end{array} \right]$ 
C) 
$\left[ \begin{array} { c c } \frac { 1 } { 6 } & - \frac { 2 } { 15 } \ 0 & - \frac { 1 } { 5 } \end{array} \right]$ 
D) 
  
A) $\left[ \begin{array} { c } 0 - \frac { 1 } { 5 } \ \frac { 1 } { 6 } - \frac { 2 } { 15 } \end{array} \right]$ 
B) $\left[ \begin{array} { c r } 0 & \frac { 1 } { 5 } \ - \frac { 1 } { 6 } & - \frac { 2 } { 15 } \end{array} \right]$ 
C) 
$\left[ \begin{array} { c c } \frac { 1 } { 6 } & - \frac { 2 } { 15 } \ 0 & - \frac { 1 } { 5 } \end{array} \right]$ 
D)

Question 19

Understand What is Meant by Equal Matrices
-$$\left[ \begin{array} { r r } 
6 & - 8 \
- 7 & 9
\end{array} \right] = \left[ \begin{array} { r r } 
x y \
- 7 & z
\end{array} \right]$$

Accepted Answer

A)  $x = 6 ; y = - 8 ; z = 9$ 
B)  $x = 6 ; y = - 8 ; z = - 7$ 
C)  $x = 6 ; y = - 7 ; z = 9$ 
D)  $x = - 8 ; y = 6 ; z = 9$ 
A)  $x = 6 ; y = - 8 ; z = 9$ 
B)  $x = 6 ; y = - 8 ; z = - 7$ 
C)  $x = 6 ; y = - 7 ; z = 9$ 
D)  $x = - 8 ; y = 6 ; z = 9$

Question 20

Inconsistent and Dependent Systems and Their Applications
1 Apply Gaussian Elimination to Systems Without Unique Solutions
-$$\begin{array} { r } 
x + 3 y + 2 z = 11 \
4 y + 9 z = - 12 \
x + 7 y + 11 z = - 1
\end{array}$$

Accepted Answer

A)  $\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } - 3 , z \right) \right\}$ 
B)  $\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}$ 
C)  $\left\{ \left( \frac { 19 z } { 4 } + 20 , \frac { 9 z } { 4 } + 3 , z \right) \right\}$ 
D)  $\left\{ \left( - \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}$ 
A)  $\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } - 3 , z \right) \right\}$ 
B)  $\left\{ \left( \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}$ 
C)  $\left\{ \left( \frac { 19 z } { 4 } + 20 , \frac { 9 z } { 4 } + 3 , z \right) \right\}$ 
D)  $\left\{ \left( - \frac { 19 z } { 4 } + 20 , - \frac { 9 z } { 4 } + 3 , z \right) \right\}$

Solve a System of Linear Equations in Three Variables Using Cramer's Rule - 3x+7z=31 5x+3y+2z=34 -9x-5y=-44

Understand What is Meant by Equal Matrices - $\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 5\end{array} \right] = \left[ \begin{array} { r r } 7 & - 3 \\7 & z\end{array} \right]$

Encode and Decode Messages -Use the coding matrix $A = \left[ \begin{array} { r r } - 1 & - 3 \\ 2 & 5 \end{array} \right]$ to encode the message CARE.

Use Determinants to Identify Inconsistent Systems and Systems with Dependent Equations - 3x+y=10 6x+2y=20

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { l l } 10 & 1 \\- 1 & 0\end{array} \right] , \quad B = \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$

Use Matrices and Gauss-Jordan Elimination to Solve Systems - -4x+7y-z=39 x+6y+9z=62 -5x+y+z=-1

Perform Scalar Multiplication -Let $A = \left[ \begin{array} { r r r } - 9 & - 3 & 8 \\ - 1 & - 6 & 8 \\ 9 & 6 & 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 3 & 8 & 9 \\ - 5 & - 4 & 1 \\ 6 & 1 & 9 \end{array} \right]$ . Find 2A - 4B.

Encode and Decode Messages -Use the coding matrix $A = \left[ \begin{array} { r r r } 1 & - 2 \\ 1 & 2 & 3 \\ 1 & 1 & 1 \end{array} \right]$ to encode the message COME_HERE.

Evaluate a Third-Order Determinant - $\left| \begin{array} { l l l } 7 & 0 & 0 \\5 & 9 & 6 \\8 & 7 & 2\end{array} \right|$

Matrix Operations and Their Applications 1 Use Matrix Notation - $\left[ \begin{array} { c c c c c } 3 & 7 & 6 & 7 & - 3 \\3 & - e & - 15 & - 13 & \pi \\- 6 & 9 & - 12 & 12 & 8 \\\frac { 1 } { 3 } & 9 & 6 & - 1 & 5\end{array} \right] ; a _ { 34 }$

Add and Subtract Matrices -A) $\left[ \begin{array} { r r } - 8 & 1 \\- 17 & 0 \\5 & - 6\end{array} \right]$

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { l l } - 4 & 6 \\ - 5 & 0 \end{array} \right]$

Understand What is Meant by Equal Matrices - $\left[ \begin{array} { r r } 6 & - 8 \\- 7 & 9\end{array} \right] = \left[ \begin{array} { r r } x y \\- 7 & z\end{array} \right]$

Inconsistent and Dependent Systems and Their Applications 1 Apply Gaussian Elimination to Systems Without Unique Solutions - x+3y+2z=11 4y+9z=-12 x+7y+11z=-1

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

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Exam 9: Matrices and Determinants