Deck 50: The Determinant of a Square Matrix

Question 1

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } - \frac { 7 } { 3 } & \frac { 14 } { 3 } \\\\- 1 & \frac { 1 } { 3 }\end{array} \right]$$ ​

A) $- \frac { 35 } { 17 }$
B) $\frac { 13 } { 35 }$
C) $\frac { 11 } { 35 }$
D) $- \frac { 17 } { 35 }$
E) $\frac { 35 } { 9 }$

Answer 1

$\frac { 35 } { 9 }$

Question 2

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the row 1.​ $$\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]$$ ​

A)-73
B)75
C)-75
D)-74
E)-76

Answer 2

-75

Question 3

Find all minors of the matrix.​ $$\left[ \begin{array} { c c c } - 7 & 0 & 4 \\2 & 4 & 3 \\3 & - 3 & 5\end{array} \right]$$ ​

A) $M _ { 11 } = 1 , M _ { 12 } = - 18 , M _ { 13 } = 12 , M _ { 21 } = - 47 , M _ { 22 } = 21 , M _ { 23 } = - 16 , M _ { 31 } = - 29 , M _ { 32 } = - 28 , M _ { 33 } = 29$
B) $M _ { 11 } = 12 , M _ { 12 } = - 47 , M _ { 13 } = 21 , M _ { 21 } = - 16 , M _ { 22 } = - 29 , M _ { 23 } = - 28 , M _ { 31 } = 29 , M _ { 32 } = 1 , M _ { 33 } = - 18$
C) $M _ { 11 } = 29 , M _ { 12 } = 1 , M _ { 13 } = - 18 , M _ { 21 } = 12 , M _ { 22 } = - 47 , M _ { 23 } = 21 , M _ { 31 } = - 16 , M _ { 32 } = - 29 , M _ { 33 } = - 28$
D) $M _ { 11 } = - 18 , M _ { 12 } = 12 , M _ { 13 } = - 47 , M _ { 21 } = 21 , M _ { 22 } = - 16 , M _ { 23 } = - 29 , M _ { 31 } = - 28 , M _ { 32 } = 29 , M _ { 33 } = 1$
E) $M _ { 11 } = - 16 , M _ { 12 } = - 29 , M _ { 13 } = - 28 , M _ { 21 } = 29 , M _ { 22 } = 1 , M _ { 23 } = - 18 , M _ { 31 } = 12 , M _ { 32 } = - 47 , M _ { 33 } = 21$

Answer 3

$M _ { 11 } = 29 , M _ { 12 } = 1 , M _ { 13 } = - 18 , M _ { 21 } = 12 , M _ { 22 } = - 47 , M _ { 23 } = 21 , M _ { 31 } = - 16 , M _ { 32 } = - 29 , M _ { 33 } = - 28$

Question 4

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } - 5 & 2 \\\frac { 1 } { 2 } & 1\end{array} \right]$$ ​

A)-6
B)-7
C)-4
D)-5
E)-8

Question 5

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 10 & 3 \\0 & 11\end{array} \right]$$ ​

A)111
B)110
C) $- 111$
D) $- 110$
E) $109$

Question 6

Find all the cofactors of the matrix.​ $$\left[ \begin{array} { c c c } - 4 & 0 & 2 \\4 & 4 & 3 \\3 & - 3 & 3\end{array} \right]$$ ​

A) $C _ { 11 } = - 24 , C _ { 12 } = 6 , C _ { 13 } = - 18 , C _ { 21 } = - 12 , C _ { 22 } = - 8 , C _ { 23 } = 20 , C _ { 31 } = - 16 , C _ { 32 } = 21 , C _ { 33 } = - 3$
B) $C _ { 11 } = - 8 , C _ { 12 } = 20 , C _ { 13 } = - 16 , C _ { 21 } = 21 , C _ { 22 } = - 3 , C _ { 23 } = - 24 , C _ { 31 } = 6 , C _ { 32 } = - 18 , C _ { 33 } = - 12$
C) $C _ { 11 } = - 3 , C _ { 12 } = - 24 , C _ { 13 } = 6 , C _ { 21 } = - 18 , C _ { 22 } = - 12 , C _ { 23 } = - 8 , C _ { 31 } = 20 , C _ { 32 } = - 16 , C _ { 33 } = 21$
D) $C _ { 11 } = 21 , C _ { 12 } = - 3 , C _ { 13 } = - 24 , C _ { 21 } = - 6 , C _ { 22 } = - 18 , C _ { 23 } = - 12 , C _ { 31 } = - 8 , C _ { 32 } = 20 , C _ { 33 } = - 16$
E) $C _ { 11 } = 6 , C _ { 12 } = - 18 , C _ { 13 } = - 12 , C _ { 21 } = - 8 , C _ { 22 } = 20 , C _ { 23 } = - 16 , C _ { 31 } = 21 , C _ { 32 } = - 3 , C _ { 33 } = - 24$

Question 7

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \\\\- 24 & \frac { 1 } { 3 }\end{array} \right]$$ ​

A) $\frac { 47 } { 6 }$
B) $- \frac { 6 } { 47 }$
C) $- \frac { 47 } { 6 }$
D) $\frac { 3 } { 47 }$
E) $\frac { 6 } { 47 }$

Question 8

Find the determinant of the matrix.​ $[ - 15 ]$ ​

A) $15$
B) $- \frac { 1 } { 15 }$
C) $- 15$
D)0
E) $\frac { 1 } { 15 }$

Question 9

Find all minors of the matrix.​ $$\left[ \begin{array} { c c } 4 & 5 \\4 & - 8\end{array} \right]$$ ​

A) $M _ { 11 } = - 8 , M _ { 12 } = - 4 , M _ { 21 } = - 5 , M _ { 22 } = - 4$
B) $M _ { 11 } = 5 , M _ { 12 } = 4 , M _ { 21 } = 8 , M _ { 22 } = 4$
C) $M _ { 11 } = 4 , M _ { 12 } = 8 , M _ { 21 } = 4 , M _ { 22 } = 5$
D) $M _ { 11 } = - 8 , M _ { 12 } = 4 , M _ { 21 } = 5 , M _ { 22 } = 4$
E) $M _ { 11 } = 4 , M _ { 12 } = 5 , M _ { 21 } = 4 , M _ { 22 } = 8$

Question 10

Find all the cofactors of the matrix.​ $$\left[ \begin{array} { c c } 6 & 7 \\4 & - 8\end{array} \right]$$ ​

A) $C _ { 11 } = - 7 , C _ { 12 } = - 4 , C _ { 21 } = - 6 , C _ { 22 } = 8$
B) $C _ { 11 } = - 6 , C _ { 12 } = - 4 , C _ { 21 } = - 7 , C _ { 22 } = 8$
C) $C _ { 11 } = - 8 , C _ { 12 } = - 4 , C _ { 21 } = - 7 , C _ { 22 } = 6$
D) $C _ { 11 } = 8 , C _ { 12 } = 4 , C _ { 21 } = 7 , C _ { 22 } = - 6$
E) $C _ { 11 } = - 6 , C _ { 12 } = - 7 , C _ { 21 } = - 4 , C _ { 22 } = 8$

Question 11

Find all the cofactors of the matrix.​ $$\left[ \begin{array} { c c c } - 5 & 0 & 3 \\4 & 3 & 6 \\6 & - 6 & 6\end{array} \right]$$ ​

A) $$C _ { 11 } = 12 , C _ { 12 } = - 42 , C _ { 13 } = - 18 , C _ { 21 } = - 48 , C _ { 22 } = - 30 , C _ { 23 } = - 9 , C _ { 31 } = 42 , C _ { 32 } = - 15 , C _ { 33 } = 54$$
B) $C _ { 11 } = - 18 , C _ { 12 } = - 48 , C _ { 13 } = - 30 , C _ { 21 } = - 9 , C _ { 22 } = 42 , C _ { 23 } = - 15 , C _ { 31 } = 54 , C _ { 32 } = 12 , C _ { 33 } = - 42$
C) $C _ { 11 } = - 9 , C _ { 12 } = 42 , C _ { 13 } = - 15 , C _ { 21 } = 54 , C _ { 22 } = 12 , C _ { 23 } = - 42 , C _ { 31 } = - 18 , C _ { 32 } = - 48 , C _ { 33 } = - 30$
D) $C _ { 11 } = - 42 , C _ { 12 } = - 18 , C _ { 13 } = - 48 , C _ { 21 } = - 30 , C _ { 22 } = - 9 , C _ { 23 } = 42 , C _ { 31 } = - 15 , C _ { 32 } = 54 , C _ { 33 } = 12$
E) $C _ { 11 } = 54 , C _ { 12 } = 12 , C _ { 13 } = - 42 , C _ { 21 } = - 18 , C _ { 22 } = - 48 , C _ { 23 } = - 30 , C _ { 31 } = - 9 , C _ { 32 } = 42 , C _ { 33 } = - 15$

Question 12

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } - 4 & 0 \\5 & 8\end{array} \right]$$ ​

A)32
B) $- 32$
C) $- \frac { 1 } { 32 }$
D)0
E) $\frac { 1 } { 32 }$

Question 13

Find all minors of the matrix.​ $$\left[ \begin{array} { c c c } - 9 & 0 & 6 \\4 & 5 & 8 \\8 & - 8 & 4\end{array} \right]$$ ​

A) $M _ { 11 } = 84 , M _ { 12 } = - 48 , M _ { 13 } = - 72 , M _ { 21 } = 48 , M _ { 22 } = - 84 , M _ { 23 } = 72 , M _ { 31 } = - 30 , M _ { 32 } = - 96 , M _ { 33 } = - 45$
B) $M _ { 11 } = - 72 , M _ { 12 } = 48 , M _ { 13 } = - 84 , M _ { 21 } = 72 , M _ { 22 } = - 30 , M _ { 23 } = - 96 , M _ { 31 } = - 45 , M _ { 32 } = 84 , M _ { 33 } = - 48$
C) $M _ { 11 } = 48 , M _ { 12 } = - 84 , M _ { 13 } = 72 , M _ { 21 } = - 30 , M _ { 22 } = - 96 , M _ { 23 } = - 45 , M _ { 31 } = 84 , M _ { 32 } = - 48 , M _ { 33 } = - 72$
D) $M _ { 11 } = - 48 , M _ { 12 } = - 72 , M _ { 13 } = 48 , M _ { 21 } = - 84 , M _ { 22 } = 72 , M _ { 23 } = - 30 , M _ { 31 } = - 96 , M _ { 32 } = - 45 , M _ { 33 } = 84$
E) $M _ { 11 } = - 30 , M _ { 12 } = - 96 , M _ { 13 } = - 45 , M _ { 21 } = 84 , M _ { 22 } = - 48 , M _ { 23 } = - 72 , M _ { 31 } = 48 , M _ { 32 } = - 84 , M _ { 33 } = 72$

Question 14

Find all the cofactors of the matrix.​ $$\left[ \begin{array} { c c } 0 & 6 \\3 & - 6\end{array} \right]$$ ​

A) $C _ { 11 } = - 6 , C _ { 12 } = - 3 , C _ { 21 } = - 6 , C _ { 22 } = 0$
B) $C _ { 11 } = 6 , C _ { 12 } = 3 , C _ { 21 } = 6 , C _ { 22 } = 0$
C) $C _ { 11 } = - 6 , C _ { 12 } = - 3 , C _ { 21 } = 0 , C _ { 22 } = 6$
D) $C _ { 11 } = 0 , C _ { 12 } = - 3 , C _ { 21 } = - 6 , C _ { 22 } = 6$
E) $C _ { 11 } = 0 , C _ { 12 } = - 6 , C _ { 21 } = - 3 , C _ { 22 } = 6$

Question 15

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 7 & 1 \\- 1 & 2\end{array} \right]$$ ​

A)15
B)13
C) $- 13$
D) $- \frac { 1 } { 13 }$
E) $- 15$

Question 16

Find all minors of the matrix.​ $$\left[ \begin{array} { c c } 0 & 8 \\4 & - 8\end{array} \right]$$ ​

A) $M _ { 11 } = 4 , M _ { 12 } = 8 , M _ { 21 } = 0 , M _ { 22 } = 8$
B) $$M _ { 11 } = - 8 , M _ { 12 } = 4 , M _ { 21 } = 8 , M _ { 22 } = 0$$
C) $$M _ { 11 } = 0 , M _ { 12 } = 8 , M _ { 21 } = 4 , M _ { 22 } = 8$$
D) $M _ { 11 } = 8 , M _ { 12 } = 4 , M _ { 21 } = 8 , M _ { 22 } = 0$
E) $M _ { 11 } = - 8 , M _ { 12 } = - 4 , M _ { 21 } = - 8 , M _ { 22 } = 0$

Question 17

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 2 & - 2 \\8 & - 16\end{array} \right]$$ ​

A) $- \frac { 1 } { 16 }$
B) $\frac { 1 } { 16 }$
C)0
D)16
E) $- 16$

Question 18

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 2 & - 5 \\0 & 0\end{array} \right]$$ ​

A) $7$
B)0
C) $- 10$
D)10
E) $- 7$

Question 19

Find the determinant of the matrix.​ $\left[ \begin{array} { c c } 0 & 5 \\- 10 & 11\end{array} \right]$ ​

A) $- 50$
B)50
C)51
D) $- 51$
E) $49$

Question 20

Find the determinant of the matrix.​ $[ 6 ]$ ​

A)0
B) $- 6$
C) $- \frac { 1 } { 6 }$
D) $6$
E) $\frac { 1 } { 6 }$

Question 21

Solve for x.​ $$\left| \begin{array} { l l } x & 3 \\1 & x\end{array} \right| = 6$$ ​

A) $x = 3$
B) $x = \pm 2$
C) $x = \pm 3$
D) $x = - 4$
E) $x = \pm 4$

Question 22

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $$\left[ \begin{array} { c c c } 16 & 24 & 8 \\0 & 40 & - 16 \\0 & 0 & - 16\end{array} \right]$$ .

A)-1,024
B)0
C) $- 10,240$
D)-2,560
E) $15,360$

Question 23

Find $A B$ .​ $A = \left[ \begin{array} { c c } - 2 & 0 \\0 & 4\end{array} \right] , B = \left[ \begin{array} { c c } 3 & 0 \\0 & - 2\end{array} \right]$ ​

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

Question 24

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $$\left[ \begin{array} { c c c } 0.2 & - 0.6 & 0.6 \\0.8 & 0.4 & 0 \\- 0.4 & - 1.4 & 1.0\end{array} \right]$$ .

A)-0.008
B)-0.080
C)-0.040
D)0.024
E)-0.016

Question 25

Given $A = \left[ \begin{array} { c c c } 0 & 2 & 4 \\- 6 & - 4 & 2 \\0 & 8 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 6 & - 4 & 0 \\2 & - 2 & 4 \\6 & 2 & 2\end{array} \right]$ ,find $| B A |$ .

A)-16,000
B)-13,440
C)19,200
D)24,000
E)25,536

Question 26

Evaluate the determinant in which the entries are functions.​ $$\left| \begin{array} { c c } x & \ln x \\1 & \frac { 6 } { x }\end{array} \right|$$ ​

A) $x - \ln x$
B) $6 x + \ln x$
C) $6 + \ln x$
D) $6 x - \ln x$
E) $6 - \ln x$

Question 27

Evaluate the determinant.​ $$\left| \begin{array} { l l l } 2 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & 6\end{array} \right|$$ ​

A)46
B)48
C)50
D)47
E)49

Question 28

Find the determinant of the matrix $$\left[ \begin{array} { l l } - 1 & 2 \\- 6 & 4\end{array} \right]$$ .

A)22
B)10
C)16
D)8
E)-2

Question 29

Find $| B |$ .​ $B = \left[ \begin{array} { c c } 5 & 0 \\0 & - 4\end{array} \right]$ ​

A)-19
B)-21
C)-22
D)-20
E)-18

Question 30

Use the matrix capabilities of a graphing utility to evaluate the determinant.​ $$\left[ \begin{array} { c c c } 3 & 1 & - 6 \\0 & - 6 & 9 \\1 & 1 & 1\end{array} \right]$$ ​

A)-72
B)-71
C)-70
D)-74
E)-73

Question 31

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2.​ $$\left[ \begin{array} { c c c } - 3 & 2 & 1 \\4 & 5 & 6 \\2 & - 3 & 1\end{array} \right]$$ ​

A)-75
B)75
C)-73
D)-74
E)-76

Question 32

Find $| A |$ .​ $A = \left[ \begin{array} { c c } - 4 & 0 \\0 & 6\end{array} \right]$ ​

A)-25
B)-26
C)-24
D)-23
E)-22

Question 33

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $$\left[ \begin{array} { c c c c c } - 1 & 1 & 0 & - 2 & 3 \\0 & 0 & 1 & 4 & 0 \\0 & - 2 & - 3 & - 5 & 1 \\1 & 0 & 0 & 7 & 0 \\0 & 0 & 0 & 1 & 0\end{array} \right]$$ .

A)-12
B)-10
C)7
D)-6
E)4

Question 34

Find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest.​ $$\left[ \begin{array} { c c c } 9 & 8 & - 2 \\2 & 2 & 0 \\- 9 & 8 & 2\end{array} \right]$$ ​

A)-64
B)-62
C)-66
D)-65
E)-63

Question 35

Find the determinant of $$\left[ \begin{array} { c c c } 0 & 1 & 1 \\2 & - 2 & 0 \\- 1 & - 8 & - 3\end{array} \right]$$ .

A)8
B)-10
C)-15
D)-12
E)2

Question 36

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $$\left[ \begin{array} { c c c c c } - 1 & 0 & 0 & 0 & 0 \\0 & 2 & 0 & 0 & 0 \\0 & 0 & - 3 & 0 & 0 \\0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & - 2\end{array} \right]$$ .

A)-6
B)4
C)-2
D)-1
E)-12

Question 37

Find the determinant of $$\left[ \begin{array} { c c c } 6 & - 2 & 0 \\- 4 & 4 & 0 \\2 & 16 & 2\end{array} \right]$$ .

A)32
B)0
C)24
D)-56
E)-64

Question 38

​Find the determinant of the matrix $$\left[ \begin{array} { c } - \frac { 1 } { 6 } - \frac { 1 } { 6 } \\\\- 1 - \frac { 2 } { 5 }\end{array} \right]$$ . ​

A)​ $- \frac { 1 } { 8 }$
B)​ $\frac { 7 } { 30 }$
C)​ $- \frac { 1 } { 10 }$
D)​ $- \frac { 1 } { 12 }$
E)​ $\frac { 7 } { 30 }$ .

Question 39

Evaluate the determinant in which the entries are functions.​ $$\left| \begin{array} { c c } 2 u & - 4 \\- 4 & 3 v\end{array} \right|$$ ​

A) $6 + 16 \text{uv}$
B) $6 \text{uv} + 16$
C) $6 \text{uv} - 16$
D) $6 - 16 \text{uv}$
E) $6 - \text{uv}$

Question 40

Find $| A B |$ .​ $A = \left[ \begin{array} { c c } - 2 & 0 \\0 & 5\end{array} \right] , B = \left[ \begin{array} { c c } 4 & 0 \\0 & - 2\end{array} \right]$ ​

A)80
B)78
C)79
D)82
E)81

Question 41

Evaluate the determinant.​

Question 42

Find all (a)minors and (b)cofactors of the matrix .Show your work.

Question 43

Find all cofactors of the matrix $$\left[ \begin{array} { c c c } - 9 & 6 & - 24 \\- 9 & - 6 & 18 \\3 & - 9 & - 18\end{array} \right]$$ . ​

A) $$\begin{array} { l l l } C _ { 11 } = 270 & C _ { 12 } = - 108 & C _ { 13 } = 99 \\C _ { 21 } = 324 & C _ { 22 } = 234 & C _ { 23 } = - 63 \\C _ { 31 } = - 36 & C _ { 32 } = 378 & C _ { 33 } = 108\end{array}$$
B) $$\begin{array} { l l l } C _ { 11 } = 270 & C _ { 12 } = 99 & C _ { 13 } = - 108 \\C _ { 21 } = 324 & C _ { 22 } = - 63 & C _ { 23 } = 234 \\C _ { 31 } = - 36 & C _ { 32 } = 108 & C _ { 33 } = 378\end{array}$$
C) $$\begin{array} { l l l l } C _ { 11 } = - 108 & C _ { 12 } = 270 & C _ { 13 } = 99 \\C _ { 21 } = 234 & C _ { 22 } = 324 & C _ { 23 } = - 63 \\C _ { 31 } = 378 & C _ { 32 } = - 36 & C _ { 33 } = 108\end{array}$$
D) $$\begin{array} { l l l } C _ { 11 } = - 36 & C _ { 12 } = 378 & C _ { 13 } = 108 \\C _ { 21 } = 324 & C _ { 22 } = 234 & C _ { 23 } = - 63 \\C _ { 31 } = 270 & C _ { 32 } = - 108 & C _ { 33 } = 99\end{array}$$
E) $$\begin{array} { l l l } C _ { 11 } = 324 & C _ { 12 } = 234 & C _ { 13 } = - 63 \\C _ { 21 } = 270 & C _ { 22 } = - 108 & C _ { 23 } = 99 \\C _ { 31 } = - 36 & C _ { 32 } = 378 & C _ { 33 } = 108\end{array}$$

Question 44

Solve for x given the following equation involving a determinant. $$\left| \begin{array} { c c } x - 6 & - 8 \\- 1 & x - 4\end{array} \right| = 0$$

A) $x = 2,8$
B) $x = - 2,8$
C) $x = 4,16$
D) $x = 2 , - 8$
E) $x = - 2 , - 8$

Question 45

Find the determinant of $$\left[ \begin{array} { c c c } 3 & - 2 & 8 \\- 3 & 2 & - 6 \\1 & - 3 & 6\end{array} \right]$$ by the method of expansion by cofactors using row 2. ​

A)14
B)-14
C)24
D)34
E)4

Question 46

Evaluate the determinant of the matrix below to determine which of the following makes the equation true. $\left| \begin{array} { c c } w & - 4 x \\- 9 y & z\end{array} \right| =$

A) $\left| \begin{array} { c c } - w & 4 x \\- 9 y & z\end{array} \right|$
B) $- \left | \begin{array} { c c } - z & 4 x \\- 9 y & - w \end{array} \right |$
C) $- \left | \begin{array} { c c } w & - 4 x \\- 9 y & z\end{array} \right |$
D) $- \left| \begin{array} { c c } - 9 y & z \\w & - 4 x\end{array} \right|$
E) $\left| \begin{array} { c c } - z & 4 x \\- 9 y & w\end{array} \right|$

Question 47

Find all minors of the matrix $$\left[ \begin{array} { c c c } 9 & - 6 & 24 \\- 9 & 6 & - 18 \\3 & - 9 & 18\end{array} \right]$$ . ​

A) $$\begin{array} { l l l } M _ { 11 } = 108 & M _ { 12 } = 90 & M _ { 13 } = - 63 \\M _ { 21 } = - 54 & M _ { 22 } = - 108 & M _ { 23 } = 63 \\M _ { 31 } = - 36 & M _ { 32 } = 54 & M _ { 33 } = 0\end{array}$$
B) $$\begin{array} { l l l l } M _ { 11 } & = 63 & M _ { 12 } = - 108 & M _ { 13 } = - 54 \\M _ { 21 } & = - 63 & M _ { 22 } = 90 & M _ { 23 } = 108 \\M _ { 31 } & = 0 & M _ { 32 } = 54 & M _ { 33 } = - 36\end{array}$$
C) $$\begin{array} { l l l } M _ { 11 } = - 54 & M _ { 12 } = - 108 & M _ { 13 } = 63 \\M _ { 21 } = 108 & M _ { 22 } = 90 & M _ { 23 } = - 63 \\M _ { 31 } = - 36 & M _ { 32 } = 54 & M _ { 33 } = 0\end{array}$$
D) $$\begin{array} { l l l l } M _ { 11 } = - 108 & M _ { 12 } = - 54 & M _ { 13 } = 63 \\M _ { 21 } = 90 & M _ { 22 } = 108 & M _ { 23 } = - 63 \\M _ { 31 } = 54 & M _ { 32 } = - 36 & M _ { 33 } = 0\end{array}$$
E) $$\begin{array} { l l l } M _ { 11 } = - 36 & M _ { 12 } = 54 & M _ { 13 } = 0 \\M _ { 21 } = 108 & M _ { 22 } = 90 & M _ { 23 } = - 63 \\M _ { 31 } = - 54 & M _ { 32 } = - 108 & M _ { 33 } = 63\end{array}$$

Question 48

Evaluate the determinant $$\left| \begin{array} { c c } 4 x & x \ln x \\4 & 6 + \ln x\end{array} \right|$$ in which the entries are functions.

A) $16 \ln x$
B) $4 - \ln x$
C) $6 \ln x$
D) $6 x$
E) $24 x$

Question 49

Evaluate the determinant $$\left | \begin{array} { c c } e ^ { 7 x } & e ^ { 4 x } \\7 e ^ { 7 x } & 4 e ^ { 4 x }\end{array} \right |$$ in which the entries are functions.

A) $- 3 e ^ { 11 x }$
B) $\frac { 7 } { 4 } e ^ { 28 x }$
C) $\frac { 4 } { 7 } e ^ { 11 x }$
D) $- 3 e ^ { 28 x }$
E) $- \frac { 3 } { 7 } e ^ { 11 x }$

Question 50

Given $A = \left[ \begin{array} { c c c } 0 & 2 & 4 \\- 6 & - 4 & 2 \\0 & 8 & 2\end{array} \right]$ ,find $| A |$ .

A)-168
B)-80
C)-88
D)40
E)120

Question 51

Solve for x given the following equation involving a determinant. $$\left| \begin{array} { c c } x + 2 & 5 \\- 1 & x + 8\end{array} \right| = 0$$

A) $x = - 8 , - 2$
B) $x = 6 , - 5$
C) $x = 8,2$
D) $x = - 7 , - 3$
E) $x = 7,3$

Question 52

Find the determinant of $$\left[ \begin{array} { c c c } 0 & - 2 & 0 \\- 3 & 2 & - 6 \\1 & 0 & 6\end{array} \right]$$ by the method of expansion by cofactors using column 3. ​

A)-14
B)24
C)-4
D)-24
E)-34