Deck 8: Matrices

Question 1

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$A = \left[ \begin{array} { r r } 3 & 7 \\- 5 & - 3\end{array} \right]$$

A)1 × 2; none of these
B)2 × 2; square matrix
C)2; square matrix
D)4; square matrix

Answer 1

B

Question 2

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$\mathrm { A } = \left[ \begin{array} { r r } 3 & - 4 \\5 & 0 \\- 5 & 9 \\4 & - 3\end{array} \right]$$

A)4 × 2; none of these
B)2; square matrix
C)8; none of these
D)2 × 4; none of these

Answer 2

A

Question 3

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $A = \left[ \begin{array} { r r } - 1 & 3 \\ 0 & 4 \\ 8 & - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 7 & 2 \\ 17 & 4 \\ 3 & 2 \end{array} \right]$ . Find $A - B$

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

Answer 3

D

Question 4

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { r r r } - 4 & 6 & 7 \\ 3 & - 5 & 12 \\ 7 & - 11 & 14 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 6 & 10 & - 4 \\ - 5 & 6 & - 8 \\ 3 & 11 & 7 \end{array} \right]$ . Find $\mathrm { A } - \mathrm { B }$

A) $\left[ \begin{array} { c c c } 10 & 4 & - 11 \\ - 8 & 11 & - 20 \\ - 4 & 22 & - 7 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 2 & 16 & 3 \\ - 2 & 1 & 4 \\ 10 & 0 & 21 \end{array} \right]$
C) $\left[ \begin{array} { r r r } - 10 & - 4 & 11 \\ 8 & - 11 & 20 \\ 4 & - 22 & 7 \end{array} \right]$
D) $\left[ \begin{array} { r r r } - 10 & - 4 & 3 \\ - 2 & - 11 & 4 \\ 4 & - 22 & 7 \end{array} \right]$

Question 5

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$A = \left[ \begin{array} { r r r r r } 9 & - 14 & 11 & - 6 & - 11 \\- 10 & 0 & 12 & - 3 & \pi \\- 13 & - 4 & 12 & 9 & - 6 \\\sqrt { 3 } & - 14 & - 3 & 11 & 15\end{array} \right]$$

A)4 × 5; none of these
B)5 × 4; none of these
C)4 × 4; square matrix
D)20; row matrix

Question 6

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $A = \left[ \begin{array} { r r r } 4 & - 6 & 4 \\ 8 & - 8 & - 2 \\ 1 & 6 & 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 3 & 6 & 2 \\ 1 & 0 & - 3 \\ - 2 & 8 & - 6 \end{array} \right]$ . Find $A + B$ .

A) $\left[ \begin{array} { r r r } 9 & 12 & 6 \\ 7 & - 8 & - 5 \\ - 1 & 14 & 0 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 7 & 0 & 6 \\ 7 & - 8 & 1 \\ 1 & 14 & 0 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 7 & 0 & 6 \\ 9 & - 8 & - 5 \\ - 1 & 14 & 0 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 9 & 12 & 6 \\ 9 & - 8 & 1 \\ 1 & 14 & 0 \end{array} \right]$

Question 7

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$A = \left[ \begin{array} { r r r } 5 & 5 & - 3 \\- 4 & - 7 & 1\end{array} \right]$$

A)2 × 3; none of these
B)3 × 2; none of these
C)3; row matrix
D)6; row matrix

Question 8

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $A = \left[ \begin{array} { r r r } 6 & 4 & 1 \\ 12 & 2 & - 1 \\ - 4 & 2 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 10 & - 4 & 1 \\ - 4 & 0 & 2 \\ 3 & 3 & - 2 \end{array} \right]$ . Find $A - B$ .

A) $\left[ \begin{array} { r r r } - 4 & - 8 & 0 \\ 16 & 2 & - 3 \\ - 7 & 1 & 5 \end{array} \right]$
B) $\left[ \begin{array} { r r r } - 4 & 0 & 0 \\ 16 & 2 & - 3 \\ 1 & - 1 & 1 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 16 & 0 & 2 \\ 8 & 2 & 1 \\ - 1 & 5 & 1 \end{array} \right]$
D) $\left[ \begin{array} { r r r } - 4 & 8 & 0 \\ 16 & 2 & - 3 \\ - 7 & - 1 & 5 \end{array} \right]$

Question 9

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { r r } - 4 & 1 \\ 2 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { l l } 6 & 2 \\ 1 & 2 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

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

Question 10

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
$$A = \left[ \begin{array} { r } - 6.5 \\- 1.6 \\1.3\end{array} \right]$$

A)3; row matrix
B)1; row matrix
C)1 × 3; column matrix
D)3 × 1; column matrix

Question 11

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$A = \left[ \begin{array} { r r r r } \sqrt { 3 } & - 7 & 1 & 6 \\\pi & 8 & - 1 & 4\end{array} \right]$$

A)4 × 2; none of these
B)2 × 4; none of these
C)3 × 2; row matrix
D)2 × 3; row matrix

Question 12

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 4 \\ 0 & 2 \end{array} \right]$ . Find $4 \mathrm {~A}$ .

A) $\left[ \begin{array} { l l } 1 & 8 \\ 4 & 6 \end{array} \right]$
B) $\left[ \begin{array} { r r } - 12 & 16 \\ 0 & 8 \end{array} \right]$
C) $\left[ \begin{array} { r r } - 12 & 16 \\ 0 & 2 \end{array} \right]$
D) $\left[ \begin{array} { r r } - 12 & 4 \\ 0 & 2 \end{array} \right]$

Question 13

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $A = \left[ \begin{array} { r r r } 7 & - 4 & 8 \\ - 6 & 5 & - 1 \\ 0 & 6 & - 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } - 2 & - 6 & - 1 \\ - 7 & - 4 & 3 \\ - 3 & - 9 & - 5 \end{array} \right]$ . Find $A - B$

A) $\left[ \begin{array} { r r r } 5 & - 10 & 7 \\ - 13 & 1 & 2 \\ - 3 & - 3 & - 8 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 9 & 2 & 9 \\ 1 & 9 & - 4 \\ 3 & 15 & 2 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 5 & - 10 & 7 \\ - 13 & 1 & - 8 \\ - 3 & - 3 & 2 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 9 & 2 & 9 \\ 1 & 9 & 2 \\ 3 & 15 & - 4 \end{array} \right]$

Question 14

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { l l } 2 & 3 \\ 2 & 4 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right]$ . Find $4 \mathrm {~A} + \mathrm { B }$ .

A) $\left[ \begin{array} { l l } 8 & 16 \\ 1 & 10 \end{array} \right]$
B) $\left[ \begin{array} { c c } 8 & 7 \\ 7 & 10 \end{array} \right]$
C) $\left[ \begin{array} { l l } 8 & 16 \\ 7 & 22 \end{array} \right]$
D) $\left[ \begin{array} { l l } 8 & 28 \\ 4 & 40 \end{array} \right]$

Question 15

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $A = \left[ \begin{array} { r r } - 1 & 0 \\ 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 1 \end{array} \right]$ . Find $A - B$ .

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

Question 16

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$A = \left[ \begin{array} { l l l l } 3 & - 3 & 9 & 7\end{array} \right]$$

A)4 × 1; column matrix
B)1 × 4; row matrix
C)1; square matrix
D)4; row matrix

Question 17

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$A = \left[ \begin{array} { r r r } 2 & 0 & 4 \\- 2 & 5 & - 5 \\0 & 8 & - 7\end{array} \right]$$

A)3 × 3; square matrix
B)9; none of these
C)6 × 6; square matrix
D)2 × 3; none of these

Question 18

Determine the size of matrix A and tell whether it is a square matrix, row matrix, column matrix, or none of these.
$$\mathrm { A } = [ 8 ]$$

A)0 × 0; none of these
B)1 × 1; square matrix
C)1; none of these
D)0; square matrix

Question 19

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { l l } - 9 & - 1 \\ - 4 & - 2 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l } - 3 & 5 \\ - 2 & 5 \end{array} \right]$ and $\mathrm { C } = \left[ \begin{array} { r r } 1 & 8 \\ - 3 & 7 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B } - \mathrm { C }$ .

A) $\left[ \begin{array} { l l } - 1 & 12 \\ - 9 & 10 \end{array} \right]$
B) $\left[ \begin{array} { c c } - 5 & 2 \\ - 5 & 0 \end{array} \right]$
C) $\left[ \begin{array} { r r } - 13 & - 4 \\ - 3 & - 4 \end{array} \right]$
D) $\left[ \begin{array} { r r } - 7 & - 14 \\ 1 & - 14 \end{array} \right]$

Question 20

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { c c } - 1 & - 1 \\ - 8 & - 2 \\ 8 & - 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 4 & 8 \\ - 4 & 1 \\ - 1 & - 1 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

A) $\left[ \begin{array} { r r } - 5 & 7 \\ - 12 & - 1 \\ 7 & - 2 \end{array} \right]$
B) $\left[ \begin{array} { r r } - 5 & - 2 \\ - 12 & - 1 \\ 7 & - 2 \end{array} \right]$
C) $\left[ \begin{array} { r r } 3 & - 9 \\ - 4 & - 3 \\ 9 & - 9 \end{array} \right]$
D) $\left[ \begin{array} { r r } - 5 & 7 \\ 12 & - 2 \\ 7 & 2 \end{array} \right]$

Question 21

If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is , what are
Let $\mathrm { A } = \left[ \begin{array} { r r } 2 & - 1 \\ 7 & 9 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 5 & - 3 \\ 4 & 7 \end{array} \right]$ . Find $- 2 \mathrm {~A} + 4 \mathrm {~B}$ .

A) $\left[ \begin{array} { l r } 7 & 4 \\ 11 & 13 \end{array} \right]$
B) $\left[ \begin{array} { l l } - 24 & - 18 \\ - 30 & - 34 \end{array} \right]$
C) $\left[ \begin{array} { r r } 16 & - 10 \\ 2 & 10 \end{array} \right]$
D) $\left[ \begin{array} { r r } - 3 & - 6 \\ 3 & 5 \end{array} \right]$

Question 22

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $A = \left[ \begin{array} { c c c } - 9 & 5 & - 6 \\ - 7 & 7 & - 1 \\ - 6 & 6 & - 6 \end{array} \right]$ and $B = \left[ \begin{array} { c c c } 4 & 3 & - 4 \\ 4 & 8 & 5 \\ 3 & 6 & 4 \end{array} \right]$ . Find $A B$ .

A) $\left[ \begin{array} { r r r } - 34 & - 23 & 37 \\ - 3 & 29 & 59 \\ - 18 & - 6 & 30 \end{array} \right]$
B) $\left[ \begin{array} { r r r } - 34 & - 3 & - 18 \\ - 23 & - 3 & - 6 \\ 37 & 59 & 30 \end{array} \right]$
C) $\left[ \begin{array} { c c c } - 36 & 15 & 24 \\ - 28 & 56 & - 5 \\ - 18 & 36 & - 24 \end{array} \right]$
D) $\left[ \begin{array} { r r r } - 9 & 5 & - 6 \\ - 7 & 7 & - 1 \\ - 6 & 6 & - 6 \\ 4 & 3 & - 4 \\ 4 & 8 & 5 \\ 3 & 6 & 4 \end{array} \right]$

Question 23

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $\mathrm { A } = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 4 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { l l l } 0 & - 2 & 5 \\ 1 & - 3 & 2 \end{array} \right]$ . Find $\mathrm { AB }$ .

A) $\left[ \begin{array} { r r } 3 & 4 \\ - 7 & - 18 \\ 1 & 23 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 0 & - 6 & 15 \\ 3 & - 12 & 8 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 3 & - 7 & 1 \\ 4 & - 18 & 23 \end{array} \right]$
D) not defined

Question 24

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $A = \left[ \begin{array} { r r } - 1 & 3 \\ 2 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 0 \\ - 1 & 1 \end{array} \right]$ . Find $A B$ .

A) $\left[ \begin{array} { l l } - 1 & 3 \\ - 6 & 2 \end{array} \right]$
B) $\left[ \begin{array} { l l } 3 & - 1 \\ 2 & - 6 \end{array} \right]$
C) $\left[ \begin{array} { r r } 2 & 0 \\ - 2 & 2 \end{array} \right]$
D) $\left[ \begin{array} { r r } 2 & - 6 \\ - 1 & - 1 \end{array} \right]$

Question 25

Find the indicated matrix.
$$A = \left[ \begin{array} { l l l } - 7 & 5 & 1 \\- 4 & 4 & 1\end{array} \right]$$

A) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

Question 26

Find the indicated matrix.
(3 × 2)(2 × 3)

A)not defined
B)3 × 3
C)2 × 2
D)2 × 3

Question 27

Find the indicated matrix.
Let $A = \left[ \begin{array} { r r } 0 & - 1 \\ 5 & 0 \\ 2 & 4 \end{array} \right] , B = \left[ \begin{array} { r r } - 4 & 0 \\ 1 & 1 \\ 5 & 5 \end{array} \right]$ , and $C = \left[ \begin{array} { r r r } - 2 & 1 & 5 \\ 0 & - 5 & - 1 \end{array} \right]$ . Find $C ( A - B )$

A) $\left[ \begin{array} { r r } - 19 & - 4 \\ - 17 & 6 \end{array} \right]$
B) $\left[ \begin{array} { r r } - 13 & - 4 \\ - 2 & - 3 \end{array} \right]$
C) $\left[ \begin{array} { r r } - 2 & - 3 \\ - 13 & - 4 \end{array} \right]$
D) $\left[ \begin{array} { r r } - 28 & 8 \\ 16 & 1 \end{array} \right]$

Question 28

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $\mathrm { A } = \left[ \begin{array} { r r } 3 & - 4 \\ - 2 & 5 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { r r r } 5 & - 2 & 8 \\ 1 & 0 & - 3 \end{array} \right]$ , and $\mathrm { C } = \left[ \begin{array} { r r r } 7 & - 9 & 0 \\ 3 & - 5 & 1 \\ - 1 & 6 & 2 \end{array} \right]$ . Find $\mathrm { AB } + \mathrm { BC }$ .

A) $\left[ \begin{array} { r r r } - 10 & - 19 & 12 \\ - 15 & 31 & - 25 \end{array} \right]$
В) $\left[ \begin{array} { r r r } 32 & 7 & 50 \\ 5 & - 23 & - 37 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 32 & 19 & 40 \\ - 15 & 31 & - 37 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 68 & 3 & 31 \\ 8 & - 2 & - 5 \end{array} \right]$

Question 29

Find the indicated matrix.
(3 × 4)(3 × 4)

A)3 × 4
B)3 × 3
C)4 × 3
D)not defined

Question 30

Find the indicated matrix.
(2 × 1)(1 × 1)

A)1 × 2
B)2 × 1
C)not defined
D)2 × 2

Question 31

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $A = \left[ \begin{array} { r r } 2 & - 8 \\ 1 & 6 \\ 0 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 3 & 2 \\ 1 & - 1 \\ 6 & 1 \end{array} \right]$ . Find $A B$ .

A) $\left[ \begin{array} { r r r } 9 & - 1 & 4 \\ - 24 & - 5 & 1 \\ 2 & 10 & 12 \end{array} \right]$
B) not defined
C) $\left[ \begin{array} { r r r } - 24 & 10 & 4 \\ 9 & - 5 & 12 \\ 2 & - 1 & 1 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 4 & 10 & - 24 \\ 12 & - 5 & 9 \\ 1 & - 1 & 2 \end{array} \right]$

Question 32

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \end{array} \right]$ . Find $A B$ .

A) $\left[ \begin{array} { r r } 9 & - 6 \\ - 6 & 8 \\ 3 & - 4 \end{array} \right]$
B) $\left[ \begin{array} { l l } 9 & 0 \\ 0 & 4 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 9 & - 6 & 3 \\ - 6 & 8 & - 4 \end{array} \right]$
D) not defined

Question 33

Find the indicated matrix.
(4 × 4)(4 × 4)

A)4 × 4
B)8 × 4
C)not defined
D)4 × 8

Question 34

Find the indicated matrix.
$$A = \left[ \begin{array} { l l } 8 & 3 \\3 & 8\end{array} \right]$$

A) $\mathrm { I } _ { \mathrm { m } } = \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 1 \\ 1 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { r r } - 1 & 0 \\ 0 & - 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$

Question 35

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $\mathrm { A } = \left[ \begin{array} { r r r } 6 & - 5 & 5 \\ - 3 & - 6 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } - 8 & - 1 & 6 \\ - 4 & - 8 & 6 \\ - 9 & - 3 & - 5 \end{array} \right]$ . Find $\mathrm { AB }$

A) $\left[ \begin{array} { r r r } - 48 & 5 & 30 \\ 12 & 48 & 30 \\ 27 & 18 & - 25 \end{array} \right]$
B) $\left[ \begin{array} { r r r } - 73 & 19 & - 19 \\ 3 & 36 & - 79 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 6 & - 5 & 5 \\ - 3 & - 6 & 5 \\ - 8 & - 6 & 5 \\ - 4 & - 8 & 6 \end{array} \right]$
D) $\left[ \begin{array} { r r } - 73 & 3 \\ 19 & 36 \\ - 19 & - 79 \end{array} \right]$ -9 -9 -5

Question 36

Find the indicated matrix.
Let $A = \left[ \begin{array} { r r r } 4 & - 8 & 3 \\ 4 & 2 & - 2 \\ 8 & 1 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 1 & - 5 & 2 \\ - 8 & 1 & - 3 \\ 4 & - 5 & - 2 \end{array} \right]$ . Find 2A-3B.

A) $\left[ \begin{array} { r r r } 5 & - 13 & 5 \\ - 4 & 3 & - 5 \\ 12 & - 4 & - 4 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 5 & - 4 & 12 \\ - 13 & 3 & - 4 \\ 5 & - 5 & - 4 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 9 & - 21 & 8 \\ 0 & 5 & - 7 \\ 20 & - 3 & - 6 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 5 & - 1 & 0 \\ 32 & 1 & 5 \\ 4 & 17 & 2 \end{array} \right]$

Question 37

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $\mathrm { A } = \left[ \begin{array} { r r r } 0 & - 3 & 1 \\ 5 & - 1 & 0 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 1 & 2 \\ 0 & 1 \\ 1 & - 1 \end{array} \right]$ . Find $\mathrm { AB }$ .

A) $\left[ \begin{array} { r r } 1 & 5 \\ - 4 & 9 \end{array} \right]$
B) $\left[ \begin{array} { r r } 0 & 10 \\ 0 & - 1 \\ 0 & 0 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 10 & - 5 & 1 \\ 5 & - 1 & 0 \\ - 5 & - 2 & 0 \end{array} \right]$
D) $\left[ \begin{array} { r r } 1 & - 4 \\ 5 & 9 \end{array} \right]$

Question 38

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
(1 × 1)(2 × 1)

A)2 × 2
B)not defined
C)1 × 2
D)2 × 1

Question 39

Find the indicated matrix.
(2 × 3)(3 × 2)

A)not defined
B)2 × 3
C)3 × 3
D)2 × 2

Question 40

Let (m × n)denote a matrix of size m × n. Find the size of the following product, or state that the product is not defined.
Let $A = \left[ \begin{array} { r r r } 1 & 3 & - 2 \\ 3 & 0 & 5 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \\ 0 & 5 \end{array} \right]$ . Find $A B$ .

A) $\left[ \begin{array} { r r } - 7 & - 3 \\ 25 & 9 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 3 & - 6 & 0 \\ 0 & 0 & 25 \end{array} \right]$
C) not defined
D) $\left[ \begin{array} { r r } - 3 & - 7 \\ 9 & 25 \end{array} \right]$

Question 41

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { l l } 2 & - 1 \\0 & - 3\end{array} \right]$$

A) $\left[ \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 6 } \\ 0 & - \frac { 1 } { 3 } \end{array} \right]$
B) $\left[ \begin{array} { c c } 0 & - \frac { 1 } { 3 } \\ \frac { 1 } { 2 } & - \frac { 1 } { 6 } \end{array} \right]$
C) $\left[ \begin{array} { r r } - \frac { 1 } { 3 } & - \frac { 1 } { 6 } \\ 0 & \frac { 1 } { 2 } \end{array} \right]$
D) $\left[ \begin{array} { l l } \frac { 1 } { 2 } & - \frac { 1 } { 6 } \\ 0 & - \frac { 1 } { 3 } \end{array} \right]$

Question 42

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { c c } 0 & 1 \\- 6 & - 4\end{array} \right]$$

A) $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & - \frac { 1 } { 6 } \\ 1 & 0 \end{array} \right]$
B) $\left[ \begin{array} { r r } 0 & - \frac { 1 } { 6 } \\ 1 & - \frac { 2 } { 3 } \end{array} \right]$
C) $\left[ \begin{array} { c c } 1 & 0 \\ - \frac { 2 } { 3 } & - \frac { 1 } { 6 } \end{array} \right]$
D) $\left[ \begin{array} { c c } - \frac { 2 } { 3 } & \frac { 1 } { 6 } \\ - 1 & 0 \end{array} \right]$

Question 43

Matrix A is given. Find appropriate identity matrices Im and In such that ImA = A and AIn = A.
$$A = \left[ \begin{array} { r r r } 2 & - 1 & 0 \\- 1 & 1 & - 2 \\1 & 0 & - 1\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 1 & 2 \\- 3 & - 2 & 4 \\- 1 & 1 & 1\end{array} \right]$$

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

Question 44

Matrix A is given. Find appropriate identity matrices Im and In such that ImA = A and AIn = A.
$$A = \left[ \begin{array} { c c } 10 & 1 \\- 1 & 0\end{array} \right] , B = \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$$

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

Question 45

Matrix A is given. Find appropriate identity matrices Im and In such that ImA = A and AIn = A.
$$A = \left[ \begin{array} { l l } 5 & 3 \\3 & 2\end{array} \right] , B = \left[ \begin{array} { r r } 2 & - 3 \\- 3 & 5\end{array} \right]$$

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

Question 46

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { r r } 1 & 4 \\1 & - 5\end{array} \right]$$

A) $\left[ \begin{array} { c c } \frac { 1 } { 9 } & \frac { 4 } { 9 } \\ \frac { 1 } { 9 } & - \frac { 5 } { 9 } \end{array} \right]$
B) $\left[ \begin{array} { c c } \frac { 5 } { 9 } & \frac { 4 } { 9 } \\ \frac { 1 } { 9 } & - \frac { 1 } { 9 } \end{array} \right]$
$\left[ \begin{array} { r r } - \frac { 5 } { 9 } & - \frac { 4 } { 9 } \\ - \frac { 1 } { 9 } & \frac { 1 } { 9 } \end{array} \right]$
D) $\left[ \begin{array} { c c } \frac { 5 } { 9 } & - \frac { 4 } { 9 } \\ \frac { 1 } { 9 } & \frac { 1 } { 9 } \end{array} \right]$

Question 47

Find the indicated matrix.
$$A = \left[ \begin{array} { r r r r } 3 & 0 & 1 & 5 \\- 4 & 9 & - 1 & 7\end{array} \right]$$

A) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 1 \\ 1 & 1 \end{array} \right]$

Question 48

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { c c } 5 & 5 \\- 5 & 0\end{array} \right]$$

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

Question 49

Find the indicated matrix.
$$A = \left[ \begin{array} { r r r } 3 & 0 & - 8 \\1 & 8 & - 8 \\- 1 & - 1 & - 2\end{array} \right]$$

A) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

Question 50

Matrix A is given. Find appropriate identity matrices Im and In such that ImA = A and AIn = A.
$A = \left[ \begin{array} { r r } - 2 & 4 \\ 4 & - 4 \end{array} \right] , B = \left[ \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 4 } \\ \frac { 1 } { 2 } & \frac { 1 } { 4 } \end{array} \right]$

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

Question 51

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { l l } - 4 & - 4 \\- 3 & - 6\end{array} \right]$$

A) $\left[ \begin{array} { c } \frac { 1 } { 4 } - \frac { 1 } { 3 } \\ - \frac { 1 } { 2 } \end{array} \right.$
B) $\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \\ \frac { 1 } { 4 } & - \frac { 1 } { 3 } \end{array} \right]$
C) $\left[ \begin{array} { c c } - \frac { 1 } { 3 } & \frac { 1 } { 3 } \\ \frac { 1 } { 4 } & - \frac { 1 } { 2 } \end{array} \right]$
D) $\left[ \begin{array} { l l } - \frac { 1 } { 2 } & - \frac { 1 } { 3 } \\ - \frac { 1 } { 4 } & - \frac { 1 } { 3 } \end{array} \right]$

Question 52

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]$

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

Question 53

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { r r r } 1 & 0 & 0 \\- 1 & 1 & 0 \\1 & 1 & 1\end{array} \right]$$

A) $\left[ \begin{array} { r r r } 1 & 0 & 0 \\ 1 & 1 & 0 \\ - 2 & - 1 & 1 \end{array} \right]$
B) $\left[ \begin{array} { l l l } 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right]$
C) $\left[ \begin{array} { r r r } - 1 & 0 & 0 \\ - 1 & - 1 & 0 \\ - 1 & - 1 & - 1 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 1 & - 1 & 1 \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \end{array} \right]$

Question 54

Solve the problem.
State University has a College of Arts & Sciences, a College of Business, and a College of Engineering. The percentage of students in each category are given by the following matrix. The student population is distributed by class and age as given in the following matrix. $$\left. \begin{array} { l c c } & \text { Female } & \text { Male } \\\text { Freshman } & 420 & 780 \\\text { Sophomore } & 550 & 750 \\\text { Junior } & 850 & 620 \\\text { Senior } & 630 & 480\end{array} \right]$$ How many female students are in the College of Business? How many male students are in the College of Arts & Sciences?

A)1308 students; 622 students
B)690 students; 513 students
C)520 students; 690 students
D)622 students; 1427 students

Question 55

Matrix A is given. Find appropriate identity matrices Im and In such that ImA = A and AIn = A.
$A = \left[ \begin{array} { l l } - 5 & 1 \\ - 7 & 1 \end{array} \right] , B = \left[ \begin{array} { l l } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \\ \frac { 7 } { 2 } & - \frac { 5 } { 2 } \end{array} \right]$

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

Question 56

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { r r r } 2 & - 1 & 0 \\- 1 & 1 & - 2 \\1 & 0 & - 1\end{array} \right]$$

A) $\left[ \begin{array} { r r r } 1 & - 2 & 2 \\ 3 & - 2 & - 4 \\ - 1 & - 1 & 2 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 1 & - 1 & 2 \\ - 2 & - 1 & - 4 \\ - 1 & - 1 & 1 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 1 & - 1 & 2 \\ - 3 & - 2 & 4 \\ - 1 & 1 & 1 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 1 & - 1 & 2 \\ - 3 & - 2 & - 4 \\ - 1 & - 1 & 1 \end{array} \right]$

Question 57

Find the indicated matrix.
$$A = \left[ \begin{array} { r r } 2 & - 7 \\6 & 0 \\- 6 & 9 \\7 & - 2\end{array} \right]$$

A) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
B) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
C) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l l } 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$
D) $\mathrm { I } _ { \mathrm { m } } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] , \mathrm { I } _ { \mathrm { n } } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

Question 58

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$A = \left[ \begin{array} { r r } - 5 & - 1 \\6 & 0\end{array} \right] , B = \left[ \begin{array} { c c } 0 & \frac { 1 } { 6 } \\- 1 & \frac { 5 } { 6 }\end{array} \right]$$

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

Question 59

Matrix A is given. Find appropriate identity matrices Im and In such that ImA = A and AIn = A.
The final grade for an algebra course is determined by grades on the midterm and final exam. The grades for four students and two possible grading systems are modeled by the following matrices.
Find the final course score for Student 3 for both grading System 1 and System 2.

A)System 1: 77.2; System 2: 76
B)System 1: 63.6; System 2: 94.4
C)System 1: 44.2; System 2: 53
D)System 1: 79.8; System 2: 79

Question 60

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 4 & 1 & 0 \\ 0 & 3 & 1 \end{array} \right]$

A) $\left[ \begin{array} { r r r } 1 & 3 & - 12 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 1 & 0 & 0 \\ - 4 & 1 & 0 \\ 12 & - 3 & 1 \end{array} \right]$
C) $\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 4 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 1 & 0 & 0 \\ 3 & - 1 & 0 \\ - 12 & 4 & 1 \end{array} \right]$

Question 61

Find the inverse of the invertible matrix.
$$\begin{array} { l } - 3 x + 9 y = 9 \\3 x + 2 y = 13\end{array}$$

A) $( 3,2 )$
В) $( - 3 , - 2 )$
C) $( 2,3 )$
D) $( - 2 , - 3 )$

Question 62

Use Cramer's rule to solve the linear system.
$$A = \left| \begin{array} { l l l } 1 & 2 & 3 \\2 & 5 & 5 \\1 & 2 & 3\end{array} \right|$$

A)-20
B)0
C)1
D)74

Question 63

Use Cramer's rule to solve the linear system.
$$A = \left| \begin{array} { r r r } - 2 & - 2 & 3 \\5 & 3 & - 5 \\- 4 & 5 & 5\end{array} \right|$$

A)-41
B)121
C)-31
D)41

Question 64

Use Cramer's rule to solve the linear system.
$$2 x + 2 y = 32$$
$2 x - 3 y = - 13$

A) $( - 7 , - 9 )$
В) $( - 9,7 )$
C) $( 9,7 )$
D) $( 7,9 )$

Question 65

Solve the linear system using an inverse matrix.
$$\begin{array} { l } 2 x + 2 y = 2 \\3 x + y = - 3\end{array}$$

A) $( 2 , - 3 )$
B) $( 3 , - 2 )$
C) $( - 2,3 )$
D) $( - 3 , - 2 )$

Question 66

Find the inverse of the invertible matrix.
$\begin{aligned} - 2 x - 6 y & = - 2 \\ 2 x - y & = - 5 \end{aligned}$

A) $( 2 , - 1 )$
B) $( - 2,1 )$
C) $( - 1,2 )$
D) $( 1 , - 2 )$

Question 67

Find the inverse of the invertible matrix.
$$\left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$$

A) $\left[ \begin{array} { l l l } 1 & - \frac { 1 } { 3 } & \frac { 1 } { 2 } \\ 1 & - \frac { 1 } { 3 } & \frac { 1 } { 3 } \\ \frac { 1 } { 7 } & \frac { 1 } { 2 } & \frac { 1 } { 8 } \end{array} \right]$
B) $\left[ \begin{array} { c c c } 1 & \frac { 1 } { 3 } & \frac { 1 } { 2 } \\ 1 & \frac { 1 } { 3 } & \frac { 1 } { 3 } \\ \frac { 1 } { 2 } & \frac { 1 } { 7 } & \frac { 1 } { 8 } \end{array} \right]$
C) $\left[ \begin{array} { r r r } - 3 & 10 & - 3 \\ 2 & - 4 & 1 \\ - 1 & 1 & 0 \end{array} \right]$
D) $\left[ \begin{array} { l l l } - 1 & - 3 & - 2 \\ - 1 & - 3 & - 3 \\ - 2 & - 7 & - 8 \end{array} \right]$

Question 68

Use Cramer's rule to solve the linear system.
$$A = \left| \begin{array} { l l l } 8 & 0 & 0 \\5 & 8 & 7 \\9 & 4 & 6\end{array} \right|$$

A)608
B)165
C)160
D)-160

Question 69

Find the inverse of the invertible matrix.
$$\begin{aligned}- 5 x + 3 y & = 8 \\3 x - 6 y & = - 30\end{aligned}$$

A) $( - 6 , - 2 )$
B) $( 6,2 )$
C) $( - 2 , - 6 )$
D) $( 2,6 )$

Question 70

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$\left[ \begin{array} { l l l } 1 & 0 & 8 \\ 1 & 2 & 3 \\ 2 & 5 & 3 \end{array} \right]$

A) $\left[ \begin{array} { l l l } 1 & 1 & 2 \\ 0 & 2 & 5 \\ 8 & 3 & 3 \end{array} \right]$
B) $\left[ \begin{array} { r r r } - 1 & - 1 & 2 \\ 0 & 2 & - 5 \\ 8 & 2 & 3 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 9 & - 40 & 16 \\ - 3 & 13 & - 5 \\ - 1 & 5 & - 2 \end{array} \right]$
D) $\left[ \begin{array} { r r r } - 1 & 0 & - 8 \\ - 1 & - 2 & - 3 \\ - 2 & - 5 & - 3 \end{array} \right]$

Question 71

Use Cramer's rule to solve the linear system.
$$\mathrm { A } = \left| \begin{array} { l l l } 2 & 2 & 2 \\6 & 6 & 1 \\3 & 2 & 1\end{array} \right|$$

A)10
B)-10
C)-22
D)94

Question 72

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { r r r } 3 & - 3 & 1 \\- 2 & 2 & - 1 \\- 4 & 5 & - 2\end{array} \right]$$

A) $\left[ \begin{array} { r r r } 1 & - 1 & 1 \\ - 2 & - 3 & 0 \\ 0 & - 2 & 1 \end{array} \right]$
B) $\left[ \begin{array} { r r r } 1 & - 3 & 1 \\ 0 & - 2 & 1 \\ - 2 & - 3 & 0 \end{array} \right]$
C) $\left[ \begin{array} { r r r } 1 & - 1 & 0 \\ - 2 & - 3 & 0 \\ 0 & - 2 & - 1 \end{array} \right]$
D) $\left[ \begin{array} { r r r } 1 & - 1 & 1 \\ 0 & - 2 & 1 \\ - 2 & - 3 & 0 \end{array} \right]$

Question 73

Use Cramer's rule to solve the linear system.
$$A = \left| \begin{array} { r r r } - 1 & 4 & 2 \\- 2 & - 2 & 5 \\- 3 & - 3 & - 3\end{array} \right|$$

A)15
B)105
C)-3
D)-105

Question 74

Find the inverse of the invertible matrix.
$$\begin{aligned}x + 2 y + 3 z & = 1 \\x + y + z & = 10 \\x - 2 z & = 7\end{aligned}$$

A) $( 31 , - 33,12 )$
B) $( 21 , - 32,12 )$
C) $( 49,67,28 )$
D) $( 1,0,0 )$

Question 75

Find the inverse of the invertible matrix.
$$\begin{array} { c } x + 2 y + 3 z = 1 \\x + y + z = - 4 \\2 x + 2 y + z = - 6\end{array}$$

A) $( - 5,12 , - 1 )$
B) $( 4,8 , - 6 )$
C) $( - 21 , - 31 , - 14 )$
D) $( - 11,9 , - 2 )$

Question 76

Solve the linear system using an inverse matrix.
$$\begin{array} { l } 3 x + 5 y = 34 \\4 x + 5 y = 37\end{array}$$

A) $( 3,5 )$
B) $( - 3 , - 5 )$
C) $( 5,3 )$
D) $( - 5,3 )$

Question 77

Find the inverse of the invertible matrix.
$$\begin{aligned}x + 3 y & = - 8 \\4 x + 4 y & = 2\end{aligned}$$

A) $( 3 , - 1 )$
В) $( - 3,1 )$
C) $( 1 , - 3 )$
D) $( - 1,3 )$

Question 78

Calculate $| \mathbf { A } |$
$$A = \left| \begin{array} { r r r } 1 & 1 & 4 \\2 & - 2 & 1 \\- 1 & - 4 & 1\end{array} \right|$$

A)-41
B)41
C)-39
D)-29

Question 79

Solve the linear system using an inverse matrix.
x + 2y + 3z = -1 x + y + z = -3 -x + y + 2z = 11

A)(30, -47, -16)
B)(7, 4, 0)
C)(-1, -12, -22)
D)(-9, 10, -4)

Question 80

Determine whether matrix B is the inverse of matrix A by finding the product AB.
$$\left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]$$

A) $\left[ \begin{array} { l l l } - 1 & - 1 & - 1 \\ - 2 & - 1 & - 1 \\ - 2 & - 2 & - 3 \end{array} \right]$
В) $\left[ \begin{array} { r r r } - 1 & 1 & 0 \\ 4 & - 1 & - 1 \\ - 2 & 0 & 1 \end{array} \right]$
C) $\left[ \begin{array} { c c c } 1 & - 1 & 1 \\ \frac { 1 } { 2 } & 1 & 1 \\ \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 3 } \end{array} \right]$
D) $\left[ \begin{array} { c c c } 1 & 1 & 1 \\ \frac { 1 } { 2 } & 1 & 1 \\ \frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 } \end{array} \right]$