Deck 8: Matrices and Determinants

Question 1

Use a graphing utility and Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { r } x + 2 y - z = - 5 \\2 x - 2 y - 2 z = - 4 \\- x + 3 y + 4 z = 6\end{array} \right.$$

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

Answer 1

(-1, -1, 2)

Question 2

Use a determinant and the given vertices of a triangle to find the area of the triangle.
(-5, 5), (3, 4), (4, -6)

A) $\frac { 79 } { 2 }$
B)- $\frac { 79 } { 2 }$
C) $\frac { 4 } { 19 }$
D)79
E) $\frac { 4 } { 11 }$

Answer 2

$\frac { 79 } { 2 }$

Question 3

Use Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { l } x + 2 y + 3 z = - 5 \\- 2 x + y - z = 10 \\3 x - 3 y + 2 z = - 21\end{array} \right.$$

A)(-2, 3, -3)
B)(7, -3, 2)
C)(2, 3, -3)
D)(2, -3, -3)
E)(-2, -3, -3)

Answer 3

(-2, 3, -3)

Question 4

​Find a value of y such that the triangle with the given vertices has an area of 4 square units. ​
​(-1, 8), ​(0, 4), ​(-1, y)
​

A)​​y = 16 or y = -8
B)​​y = 16 or y = 0
C)​y = -8 or y = 0
D)​y = -16 or y = 0
E)​y = 8 or y = 0

Question 5

Use a determinant to find an equation of the line passing through the points. ​
(0, 0), (3, 5)
​

A)3x - 5y = 0
B)5x - 3y = 5
C)5x - 3y = 0
D)5x + 3y = 3
E)5x + 3y = 0

Question 6

Find y such that the points are collinear. ​
(-3, 3), (-4, y), (-2, 4)
​

A)y = 3
B)y = 4
C)y = -2
D)y = 2
E)y = 5

Question 7

Use Cramer's Rule to solve (if possible) the system of equations.
$$\left\{ \begin{array} { r } 5 x - 4 y + z = - 12 \\- x + 2 y - 2 z = 12 \\3 x + y + z = - 2\end{array} \right.$$

A)(0, 2, -4)
B)(0, -2, 4)
C)(0, -2, -4)
D)(0, 2, 0)
E)(0, 2, 4)

Question 8

Find y such that the points are collinear. ​
(5, -4), (3, y), (4, -3)
​

A)y = -2
B)y = 1
C)y = 0
D)y = -1
E)y = 2

Question 9

Use a determinant and the given vertices of a triangle to find the area of the triangle.
(-2, 4), (2, 5), (6, -4)

A) $\frac { 6 } { 19 }$
B)20
C) $\frac { 5 } { 17 }$
D)-20
E)40

Question 10

Use Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { r } - 7 x + 11 y = 20 \\3 x - 9 y = 0\end{array} \right.$$

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

Question 11

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 1.6 \\0.2 x + 0.3 y = 2.3\end{array} \right.$

A) $\left( - \frac { 34 } { 7 } , - \frac { 31 } { 7 } \right)$
B) $\left( \frac { 34 } { 7 } , \frac { 31 } { 7 } \right)$
C) $\left( \frac { 34 } { 7 } , - \frac { 31 } { 7 } \right)$
D) $\left( \frac { 31 } { 7 } , \frac { 34 } { 7 } \right)$
E) $\left( - \frac { 34 } { 7 } , \frac { 31 } { 7 } \right)$

Question 12

Use Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { l } 4 x - 2 y + z = - 3 \\2 x + 2 y + 3 z = 9 \\5 x - 2 y + 6 z = - 15\end{array} \right.$$

A)(3, -6, -3)
B)(3, 6, -3)
C)(-3, 6, -3)
D)(-3, -6, -3)
E)(7, -6, -3)

Question 13

Use Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 5.52 \\0.2 x + 0.3 y = 2.63\end{array} \right.$$

A) $\left( \frac { 77 } { 10 } , \frac { 8 } { 5 } \right)$
B) $\left( \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$
C) $\left( - \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$
D) $\left( \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$
E) $\left( - \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$

Question 14

Use a determinant and the given vertices of a triangle to find the area of the triangle.
(0, -3), (-3, 3), (4, 4)

A) $\frac { 4 } { 7 }$
B) $\frac { 5 } { 11 }$
C) $\frac { 45 } { 2 }$
D) $\frac { 4 } { 19 }$
E)45

Question 15

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } 6 x - 5 y = 23 \\- 13 x + 3 y = - 89\end{array} \right.$

A)(-8, -5)
B)(8, -5)
C)(-8, 5)
D)(8, 5)
E)(5, 8)

Question 16

Use a determinant and the given vertices of a triangle to find the area of the triangle.
(-2, 5), (3, 3), (-1, 6)

A) $\frac { 7 } { 2 }$
B) $\frac { 2 } { 19 }$
C) $\frac { 3 } { 7 }$
D) $\frac { 6 } { 11 }$
E) $\frac { 5 } { 17 }$

Question 17

Use a determinant to find an equation of the line passing through the points. ​
(-5, 3), (1, 1)
​

A)2x -6y + 8 = 0
B)2x + 6y + 8 = 0
C)2x -6y - 8 = 0
D)2x + 6y - 8 = 0
E)2x + 6y - 8 = 5

Question 18

Use a determinant to find an equation of the line passing through the points. ​
(8, 8), (-4, -8)
​

A)4x + 3y + 8 = 0
B)4x + 3y - 8 = 0
C)4x - 3y + 8 = 0
D)4x - 3y - 8 = 0
E)4x - 3y = 0

Question 19

Use Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { l } 4 x - 3 y = - 30 \\6 x + 9 y = - 18\end{array} \right.$$

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

Question 20

Use Cramer's Rule to solve (if possible) the system of equations. $$\left\{ \begin{array} { r } 4 x - y + z = - 20 \\2 x + 2 y + 3 z = 10 \\5 x - 2 y + 6 z = 2\end{array} \right.$$

A)(6, 2, 6)
B)(6, -2, 6)
C)(-6, 2, 6)
D)(7, -2, 6)
E)(-6, -2, 6)

Question 21

​Find a value of y such that the triangle with the given vertices has an area of 26 square units. ​
​(-6, -6), ​(2, -2), ​(-9, y)
​

A)​y = -1 or y = -14
B)​y = -1 or y = 14
C)​y = -1 or y = 0
D)​y = 1 or y = 14
E)​y = 1 or y = -14

Question 22

A triangular region of farmland has become overrun with deer and will be open to the public for hunting to reduce the population.In order to know how many hunters to allow on the land at one time, you need to know the area of the region in square miles.In order to estimate the area of the region, you travel from the southernmost vertex C north 20 miles then west 24 miles (for vertex B), and from the southernmost vertex C you travel 54 miles north then 7 miles east (for vertex A).Use a graphing utility to approximate the number of square miles of land.

A)578 square miles
B)51 square miles
C)119 square miles
D)429 square miles
E)718 square miles

Question 23

Use Cramer's Rule to solve the following system of linear equations: $$\left\{ \begin{array} { r } 4 x - 8 z = 2 \\- 4 y + 12 z = 3 \\8 x + 20 z = 0\end{array} \right.$$

A) $x = \frac { 1 } { 2 } ; y = - \frac { 5 } { 4 } ; z = - \frac { 1 } { 2 }$
B) $x = \frac { 5 } { 18 } ; y = - \frac { 13 } { 12 } ; z = - \frac { 1 } { 9 }$
C) $x = \frac { 1 } { 18 } ; y = - \frac { 13 } { 12 } ; z = - \frac { 1 } { 18 }$
D) $x = \frac { 5 } { 18 } ; y = - \frac { 11 } { 12 } ; z = - \frac { 1 } { 2 }$
E) $x = \frac { 1 } { 2 } ; y = - \frac { 5 } { 4 } ; z = - \frac { 1 } { 2 }$ .

Question 24

Use a determinant to find y such that (8, -20), (16, y), and (20, -8) are collinear.

A)y = -12
B)y = 20
C)y = -44
D)y = 4
E)y = -8

Question 25

Find a value of y such that the triangle with the given vertices has an area of 8 square units. ​
​(5, 6), ​(5, 8), ​(-3, y)
​

A)y = 10 or y = -9
B)y = -10 or y = -9
C)y = 10 or y = 9
D)y = 10 or y = 0
E)y = -10 or y = 9

Question 26

Use a determinant and the given vertices of a triangle to find the area of the triangle.

A)27
B) $\frac { 4 } { 19 }$
C)29
D) $\frac { 25 } { 2 }$
E) $\frac { 6 } { 11 }$

Question 27

Use a determinant to determine whether the points (-5, 1), (-8, -1) and (1, 5) are collinear.

A) $\left| \begin{array} { c c c } - 5 & 1 & 1 \\- 8 & - 1 & 1 \\1 & 5 & 1\end{array} \right| = 0 \text {; therefore, the points are collinear. }$
B) $\left| \begin{array} { c c c } - 5 & 1 & 1 \\- 8 & - 1 & 1 \\1 & 5 & 1\end{array} \right| = 1 \text {; therefore, the points are not collinear }$

Question 28

Use a determinant and the given vertices of a triangle to find the area of the triangle.

A) $\frac { 6 } { 7 }$
B)4
C) $\frac { 2 } { 17 }$
D) $\frac { 6 } { 11 }$
E)7

Question 29

Use a determinant to find the area of the triangle shown below.

A)14 square units
B) $\frac { 57 } { 4 }$ square units
C) $\frac { 27 } { 2 }$ square units
D)12 square units
E) $\frac { 85 } { 6 }$ square units

Question 30

Determine a positive value for y such that a triangle with vertices (-4, 2), (6, 4) and (0, y) has an area of 16 square units.

A)11
B)8
C)6
D)9
E)1

Question 31

Use a determinant and the given vertices of a triangle to find the area of the triangle.

A) $\frac { 2 } { 13 }$
B) $\frac { 6 } { 19 }$
C)18
D)38
E)40

Question 32

Use determinants to find the area of a triangle with the given vertices and confirm your answer by plotting the points in a coordinate plane and using the formula $$\text { Area } \left. = \frac { 1 } { 2 } \text { (base)(height } \right)$$ .
(5, -3), (2, -2), (7, 5)

A)21
B)13
C)15
D)19
E)17

Question 33

You inherited a triangular piece of property after your Uncle Izzy passed away.You want to know the size of land, so you "step it off" to estimate the square footage.From the southernmost vertex A, you travel north 300 feet then west 220 feet (for vertex C), and from the southernmost vertex A, you travel 420 feet north then 50 feet west (for vertex B).Use a graphing utility to approximate the number of square feet of land that you have inherited.

A)3,000 square feet
B)38,700 square feet
C)53,700 square feet
D)43,500 square feet
E)22,500 square feet

Question 34

Use Cramer's Rule to solve the following system of linear equations: $$\left\{ \begin{aligned}14 x - 21 y + 14 z & = 1 \\- 21 x + 14 y + 7 z & = 5 \\28 x + 7 y - 21 z & = 5\end{aligned} \right.$$

A) $x = \frac { 5 } { 7 } ; y = \frac { 55 } { 49 } ; z = \frac { 95 } { 147 }$
B) $x = \frac { 3 } { 7 } ; y = \frac { 45 } { 98 } ; z = \frac { 12 } { 49 }$
C) $x = \frac { 11 } { 14 } ; y = \frac { 115 } { 98 } ; z = \frac { 106 } { 147 }$
D) $x = \frac { 11 } { 21 } ; y = \frac { 115 } { 147 } ; z = \frac { 106 } { 147 }$
E) $x = \frac { 3 } { 7 } ; y = \frac { 75 } { 98 } ; z = \frac { 27 } { 49 }$

Question 35

A large region of forest has been infested with gypsy moths.The region is roughly triangular, as shown in the figure on the next page.From the northernmost vertex A of the region, the distances to the other vertices are x = 25 miles south and 10 miles east (for vertex B), and 20 miles south and 28 miles east (for vertex C).Use a graphing utility to approximate the number of square miles in this region. ​ ​

A)250 mi²
B)280 mi²
C)260 mi²
D)290 mi²
E)270 mi²

Question 36

Use a determinant and the given vertices of a triangle to find the area of the triangle.

A) $\frac { 41 } { 2 }$
B)21
C) $\frac { 39 } { 2 }$
D)20
E) $\frac { 43 } { 2 }$

Question 37

Use a determinant to find an equation of the line passing through the points. ​
(0, 0), (-8, 2)
​

A)2x + 8y = 0
B)8x - 2y = 2
C)2x + 8y = 8
D)8x - 2y = 0
E)2x - 8y = 0

Question 38

A hair product company sells three types of hair products for $30, $20 and $10 per unit.In one year, the total revenue for the three products was $820,000 which corresponded to the sale of 42,000 units.The company sold half as many units of the $30 products as units of the $20 product.Use Cramer's Rule to solve a system of linear equations to find how many units of each product were sold. ​

A)10,000 units at $30, 20,000 units at $20, 12,000 units at $10.
B)10,000 units at $30, 20,000 units at $20, 18,000 units at $10.
C)10,000 units at $30, 20,000 units at $20, 20,000 units at $10.
D)10,000 units at $30, 20,000 units at $20, 14,000 units at $10.
E)10,000 units at $30, 20,000 units at $20, 16,000 units at $10.

Question 39

Use a determinant and the given vertices of a triangle to find the area of the triangle.

A)52
B) $\frac { 4 } { 19 }$
C)25
D) $\frac { 2 } { 17 }$
E)54

Question 40

Find a value of y such that the triangle with the given vertices has an area of 40 square units. ​
​(7, 0), ​(7, -5), ​(-9, y)
​

A)​y = -9 or y = -10
B)​​y = 7 or y = -3
C)​y = 9 or y = 0
D)​y = 8 or y = 3
E)​y = 10 or y = 8

Question 41

Write a cryptogram for the message "TWO IF BY LAND" using the matrix .Show all your work.

Question 42

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix .Show all your work.
​

Question 43

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix, by assigning a number to each letter in the alphabet (with 0 assigned to a blank space, 0 = _, 1 = A, 2 = B and so on.) $\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 1 & 1 & 1 \\1 & - 5 & 0\end{array} \right]$ .

A) $\begin{array} { c c c c c c c c c } 1 & - 36 & 28 & 18 & - 29 & 14 & 13 & - 20 & 18 \\- 25 & 115 & 43 & 10 & - 67 & 11 & & &\end{array}$
B) $\begin{array} { c c c c c c c c c } 13 & - 20 & 18 & - 25 & 115 & 43 & 10 & - 67 & 11 \\1 & - 36 & 28 & 18 & - 29 & 14 & & &\end{array}$
C) $\begin{array} { r r r l l l l l l } 10 & - 67 & 11 & 1 & - 36 & 28 & 18 & - 29 & 14 \\13 & - 20 & 18 & - 25 & 115 & 43 & & &\end{array}$
D) $\begin{array} { c c c c c c c c c } - 25 & 115 & 43 & 10 & - 67 & 11 & 1 & - 36 & 28 \\18 & - 29 & 14 & 13 & - 20 & 18 & & &\end{array}$
E) $\begin{array} { c c c c c c c c c } 18 & - 29 & 14 & 13 & - 20 & 18 & - 25 & 115 & 43 \\10 & - 67 & 11 & 1 & - 36 & 28 & & &\end{array}$

Question 44

Use a determinant to find an equation of the line passing through the points (-5, -1) and (2, 4). ​

A)5x + 7y - 18 = 0
B)-5x - 7y - 18 = 0
C)-5x + 7y + 18 = 0
D)5x - 7y - 18 = 0
E)-5x + 7y - 18 = 0

Question 45

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 46

Determine a positive value for y such that a triangle with vertices $P ( 0,0 ) , Q ( 17,0 )$ , and $R ( 17 , y )$ has an area of 17 square units.

A)8
B)2
C)9
D)11
E)1

Question 47

Use a determinant to determine whether the points and are collinear.Show all work.
​

Question 48

Write a cryptogram by assigning a number to each letter in the alphabet such as 1 = A, 2 = B, 3 = C and so on (with 0 assigned to a blank space) for the message "TWO IF BY LAND" using the following matrix. $$\left[ \begin{array} { c c c } 0 & 1 & - 1 \\- 1 & - 1 & 1 \\- 1 & 1 & 0\end{array} \right]$$

A) $\begin{array} { l l l l l l l l l } - 13 & - 11 & 12 & - 4 & 10 & - 10 & - 38 & 12 & 3 \\- 15 & - 3 & 9 & - 27 & 23 & 2 & & &\end{array}$
B) $\begin{array} { l l l l l l l l l } - 4 & 10 & - 10 & - 38 & 12 & 3 & - 15 & - 3 & 9 \\- 27 & 23 & 2 & - 13 & - 11 & 12 & & &\end{array}$
C)
$\begin{array} { l l l l l l l l l } - 27 & 23 & 2 & - 13 & - 11 & 12 & - 4 & 10 & - 10 \\- 38 & 12 & 3 & - 15 & - 3 & 9 & & &\end{array}$
D) $\begin{array} { l l l l l l l l l } - 15 & - 3 & 9 & - 27 & 23 & 2 & - 13 & - 11 & 12 \\- 4 & 10 & - 10 & - 38 & 12 & 3 & & &\end{array}$
E) $\begin{array} { c c c c c c c c c } - 38 & 12 & 3 & - 15 & - 3 & 9 & - 27 & 23 & 2 \\- 13 & - 11 & 12 & - 4 & 10 & - 10 & & &\end{array}$

Question 49

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND" by assigning a number to each letter in the alphabet such as 1 = A, 2 = B, 3 = C and so on (with 0 assigned to a blank space);
Then encode the message using the encoding matrix $$\left[ \begin{array} { c c c } 0 & 3 & - 3 \\- 3 & - 3 & 3 \\- 3 & 3 & 0\end{array} \right]$$ .

A)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 20 & 23 & 15\end{array} \right] \left[ \begin{array} { l l l } 0 & 9 & 6\end{array} \right] \left[ \begin{array} { l l l } 0 & 2 & 25\end{array} \right] \left[ \begin{array} { l l l } 0 & 12 & 1\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}\end{array} \\{ \left[ \begin{array} { l l l } 14 & 4 & 0\end{array} \right] } \\\mathrm { N } \mathrm { D } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } - 114 & 36 & 9\end{array} \right] \left[ \begin{array} { l l l } - 45 & - 9 & 27\end{array} \right] \left[ \begin{array} { l l l } - 81 & 69 & 6\end{array} \right] } \\{ \left[ \begin{array} { l l l } - 39 & - 33 & 36\end{array} \right] \left[ \begin{array} { l l l } - 12 & 30 & - 30\end{array} \right] }\end{array}$$
B)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 20 & 23 & 15\end{array} \right] \left[ \begin{array} { l l l } 0 & 9 & 6\end{array} \right] \left[ \begin{array} { l l l } 0 & 2 & 25\end{array} \right] \left[ \begin{array} { l l l } 0 & 12 & 1\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}\end{array} \\{ \left[ \begin{array} { l l l } 14 & 4 & 0\end{array} \right] } \\\mathrm { N } \mathrm { D } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } - 81 & 69 & 6\end{array} \right] \left[ \begin{array} { l l l } - 39 & - 33 & 36\end{array} \right] \left[ \begin{array} { l l l } - 12 & 30 & - 30\end{array} \right] } \\{ \left[ \begin{array} { l l l l } - 114 & 36 & 9\end{array} \right] \left[ \begin{array} { l l l } - 45 & - 9 & 27\end{array} \right] }\end{array}$$
C)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 20 & 23 & 15\end{array} \right] \left[ \begin{array} { l l l } 0 & 9 & 6\end{array} \right] \left[ \begin{array} { l l l } 0 & 2 & 25\end{array} \right] \left[ \begin{array} { l l l } 0 & 12 & 1\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}\end{array} \\{ \left[ \begin{array} { l l l } 14 & 4 & 0\end{array} \right] } \\\mathrm { N } \mathrm { D } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } - 39 & - 33 & 36\end{array} \right] \left[ \begin{array} { l l l } - 12 & 30 & - 30\end{array} \right] \left[ \begin{array} { l l l l } - 114 & 36 & 9\end{array} \right] } \\{ \left[ \begin{array} { l l l } - 45 & - 9 & 27\end{array} \right] \left[ \begin{array} { l l l } - 81 & 69 & 6\end{array} \right] }\end{array}$$
D)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 20 & 23 & 15\end{array} \right] \left[ \begin{array} { l l l } 0 & 9 & 6\end{array} \right] \left[ \begin{array} { l l l } 0 & 2 & 25\end{array} \right] \left[ \begin{array} { l l l } 0 & 12 & 1\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}\end{array} \\{ \left[ \begin{array} { l l l } 14 & 4 & 0\end{array} \right] } \\\mathrm { N } \mathrm { D } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } - 12 & 30 & - 30\end{array} \right] \left[ \begin{array} { l l l l } - 114 & 36 & 9\end{array} \right] \left[ \begin{array} { l l l } - 45 & - 9 & 27\end{array} \right] } \\{ \left[ \begin{array} { l l l } - 81 & 69 & 6\end{array} \right] \left[ \begin{array} { l l l l } - 39 & - 33 & 36\end{array} \right] }\end{array}$$
E)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 20 & 23 & 15\end{array} \right] \left[ \begin{array} { l l l } 0 & 9 & 6\end{array} \right] \left[ \begin{array} { l l l } 0 & 2 & 25\end{array} \right] \left[ \begin{array} { l l l } 0 & 12 & 1\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } T & \text { W } & \text { O } &&& \text { IF } &&&& \text { B Y } &&& \text { L A}\end{array} \\{ \left[ \begin{array} { l l l } 14 & 4 & 0\end{array} \right] } \\\mathrm { N } \mathrm { D } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } - 45 & - 9 & 27\end{array} \right] \left[ \begin{array} { l l l } - 81 & 69 & 6\end{array} \right] \left[ \begin{array} { l l l } - 39 & - 33 & 36\end{array} \right] } \\{ \left[ \begin{array} { l l l } - 12 & 30 & - 30\end{array} \right] \left[ \begin{array} { l l l } - 114 & 36 & 9\end{array} \right] }\end{array}$$

Question 50

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS" by assigning a number to each letter in the alphabet such as 1 = A, 2 = B, 3 = C and so on (with 0 assigned to a blank space);
Then encode the message using the encoding matrix $$\left[ \begin{array} { c c c } 0 & - 2 & 1 \\- 1 & 1 & 1 \\1 & 2 & 0\end{array} \right]$$ .

A)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 13 & 5 & 18\end{array} \right] \left[ \begin{array} { l l l } 18 & 25 & 0\end{array} \right] \left[ \begin{array} { l l l } 3 & 8 & 18\end{array} \right] \left[ \begin{array} { l l l } 9 & 19 & 20\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } M & \text { E } & \text { R } && \text { R Y } &&&&& \text { C R I } &&& \text { S T M}\end{array} \\{ \left[ \begin{array} { l l l } 13 & 1 & 19\end{array} \right] } \\\text { M A S } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 13 & 15 & 18\end{array} \right] \left[ \begin{array} { l l l } - 25 & - 11 & 43\end{array} \right] \left[ \begin{array} { l l l } 10 & 38 & 11\end{array} \right] } \\{ \left[ \begin{array} { l l l } 1 & 41 & 28\end{array} \right] \left[ \begin{array} { l l l } 18 & 13 & 14\end{array} \right] }\end{array}$$
B)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 13 & 5 & 18\end{array} \right] \left[ \begin{array} { l l l } 18 & 25 & 0\end{array} \right] \left[ \begin{array} { l l l } 3 & 8 & 18\end{array} \right] \left[ \begin{array} { l l l } 9 & 19 & 20\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } M & \text { E } & \text { R } & \text { } & \text { R Y } &&&& \text { CR I } & & &\text { S } & \text { T }& \text { M }\end{array} \\{ \left[ \begin{array} { l l l } 13 & 1 & 19\end{array} \right] } \\\text { M A S } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 18 & 13 & 14\end{array} \right] \left[ \begin{array} { l l l } 13 & 15 & 18\end{array} \right] \left[ \begin{array} { l l l } - 25 & - 11 & 43\end{array} \right] } \\{ \left[ \begin{array} { l l l } 10 & 38 & 11\end{array} \right] \left[ \begin{array} { l c l } 1 & 41 & 28\end{array} \right] }\end{array}$$
C)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 13 & 5 & 18\end{array} \right] \left[ \begin{array} { l l l } 18 & 25 & 0\end{array} \right] \left[ \begin{array} { l l l } 3 & 8 & 18\end{array} \right] \left[ \begin{array} { l l l } 9 & 19 & 20\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } M & \text { E } & \text { R }&&& \text {R Y } & &&\text { C R I } &&&& \text { S T M }\end{array} \\{ \left[ \begin{array} { l l l } 13 & 1 & 19\end{array} \right] } \\\text { M A S } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } - 25 & - 11 & 43\end{array} \right] \left[ \begin{array} { l l l } 10 & 38 & 11\end{array} \right] \left[ \begin{array} { l l l } 1 & 41 & 28\end{array} \right] } \\{ \left[ \begin{array} { l l l } 18 & 13 & 14\end{array} \right] \left[ \begin{array} { l l l } 13 & 15 & 18\end{array} \right] }\end{array}$$
D)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 13 & 5 & 18\end{array} \right] \left[ \begin{array} { l l l } 18 & 25 & 0\end{array} \right] \left[ \begin{array} { l l l } 3 & 8 & 18\end{array} \right] \left[ \begin{array} { l l l } 9 & 19 & 20\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } M & \text { E } & \text { R } &&\text { R Y } &&&& \text { C R I } &&&& \text { S T M}\end{array} \\{ \left[ \begin{array} { l l l } 13 & 1 & 19\end{array} \right] } \\\text { M A S } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 10 & 38 & 11\end{array} \right] \left[ \begin{array} { l l l } 1 & 41 & 28\end{array} \right] \left[ \begin{array} { l l l } 18 & 13 & 14\end{array} \right] } \\{ \left[ \begin{array} { l l l } 13 & 15 & 18\end{array} \right] \left[ \begin{array} { l l l } - 25 & - 11 & 43\end{array} \right] }\end{array}$$
E)Uncoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 13 & 5 & 18\end{array} \right] \left[ \begin{array} { l l l } 18 & 25 & 0\end{array} \right] \left[ \begin{array} { l l l } 3 & 8 & 18\end{array} \right] \left[ \begin{array} { l l l } 9 & 19 & 20\end{array} \right] } \\\begin{array} { l l l l l l l l l l l l } M & \text { E } & \text { R } &&\text { R Y } &&&& \text { C R I } &&&& \text { S T M}\end{array} \\{ \left[ \begin{array} { l l l } 13 & 1 & 19\end{array} \right] } \\\text { M A S } \\\end{array}$$ Encoded: $$\begin{array} { l } { \left[ \begin{array} { l l l } 1 & 41 & 28\end{array} \right] \left[ \begin{array} { l l l } 18 & 13 & 14\end{array} \right] \left[ \begin{array} { l l l } 13 & 15 & 18\end{array} \right] } \\{ \left[ \begin{array} { l l l } - 25 & - 11 & 43\end{array} \right] \left[ \begin{array} { l l l } 10 & 38 & 11\end{array} \right] } \\\end{array}$$

Question 51

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS"; then encode the message using the encoding matrix .Show all your work.

Question 52

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

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

Question 53

Use Cramer's rule to find the solution of the system, if possible. $$\left\{ \begin{aligned}5 x - y & = 12 \\x + y & = 0\end{aligned} \right.$$

A)x = 2, y = -2
B)x = 5, y = -12
C)x = 5, y = -5
D)The system is inconsistent.
E)The equations are dependent.

Question 54

Use determinants to find the area of the triangle with vertices at the given points. ​
P(0, 0), Q(4, 0), R(4, 3)
​

A)A = 12
B)A = 4
C)A = 6
D)A = 3
E)none of these

Question 55

Use a determinant to determine whether the points (-3, -9), (-5, -11) and (0, -7) are collinear.

A) $\left| \begin{array} { c c c } - 3 & - 9 & 1 \\- 5 & - 11 & 1 \\0 & - 7 & 1\end{array} \right| = 2 \text {; therefore, the points are not collinear }$
B) $\left| \begin{array} { c c c } - 3 & - 9 & 1 \\- 5 & - 11 & 1 \\0 & - 7 & 1\end{array} \right| = 0 \text {; therefore, the points are collinear }$

Question 56

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND"; then encode the message using the encoding matrix .Show all your work.

Question 57

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 58

Use Cramer's Rule to solve the following system of linear equations: $$\left\{ \begin{aligned}3 x - 6 z & = 8 \\- 6 y + 13 z & = 3 \\6 x + 15 z & = 0\end{aligned} \right.$$

A) x = $\frac { 40 } { 27 }$ , y = $- \frac { 16 } { 27 }$ , z = $- \frac { 289 } { 162 }$
B) x = $\frac { 40 } { 27 }$ , y = $- \frac { 289 } { 162 }$ , z = $- \frac { 16 } { 27 }$
C) x = $\frac { 27 } { 40 }$ , y = $- \frac { 162 } { 289 }$ , z = $- \frac { 27 } { 16 }$
D) x = $\frac { 27 } { 40 }$ , y = $- \frac { 289 } { 162 }$ , z = $- \frac { 16 } { 27 }$
E) x = $\frac { 40 } { 27 }$ , y = $- \frac { 162 } { 289 }$ , z = $- \frac { 16 } { 27 }$

Question 59

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

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

Question 60

Use a determinant to find y such that $( 6 , - 15 ) , ( 12 , y )$ , and $( 15 , - 6 )$ are collinear.

A) $y = - 33$
B) $y = 15$
C) $y = 3$
D) $y = - 6$
E) $y = - 9$

Question 61

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 62

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

Question 63

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 64

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 65

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 66

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

Question 67

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 68

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 69

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 70

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) $\frac { 35 } { 9 }$

Question 71

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 72

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 73

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 74

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 75

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 76

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 77

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 78

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 79

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 80

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