Deck 49: The Inverse of a Square Matrix

Question 1

Find the inverse of the matrix (if it exists). ​​ $$\left[ \begin{array} { l l } 1 & 2 \\3 & 7\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { l l } 7 & 2 \\3 & 1\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c } 7 & - 2 \\- 3 & 1\end{array} \right]$$
C)​ $$\left[ \begin{array} { l l } - 7 & - 2 \\- 3 & - 1\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c } - 7 & - 2 \\3 & 1\end{array} \right]$$
E)​ $$\left[ \begin{array} { c c } - 7 & 2 \\3 & - 1\end{array} \right]$$

Answer 1

​ $$\left[ \begin{array} { c c } 7 & - 2 \\- 3 & 1\end{array} \right]$$

Question 2

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists). ​​ $$\left[ \begin{array} { l l } 2 & 3 \\1 & 5\end{array} \right]$$ ​

A)​ $$\frac { 1 } { 7 } \left[ \begin{array} { l l } 5 & 3 \\1 & 2\end{array} \right]$$
B)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } - 5 & - 3 \\1 & 2\end{array} \right]$$
C)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } - 5 & 3 \\1 & - 2\end{array} \right]$$
D)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 5 & - 3 \\- 1 & 2\end{array} \right]$$
E)​ $$\frac { 1 } { 7 } \left[ \begin{array} { l l } - 5 & - 3 \\- 1 & - 2\end{array} \right]$$

Answer 2

​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 5 & - 3 \\- 1 & 2\end{array} \right]$$

Question 3

Solve the system of linear equations. ​​ $$\left\{ \begin{array} { l } x - 2 y = 0 \\2 x - 3 y = 8\end{array} \right.$$ ​

A)​ $( 16,8 )$
B)​ $( 16 , - 8 )$
C)​ $( - 8,16 )$
D)​ $( 8,16 )$
E)​ $( - 16,8 )$

Answer 3

​ $( 16,8 )$

Question 4

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists).​ $$\left[ \begin{array} { c c } 6 & 7 \\- 5 & 13\end{array} \right]$$ ​

A)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 13 & - 7 \\5 & 6\end{array} \right]$$
B)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } - 13 & 7 \\5 & - 6\end{array} \right]$$
C)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } - 13 & - 7 \\5 & 6\end{array} \right]$$
D)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } 13 & 7 \\5 & 6\end{array} \right]$$
E)​ $$\frac { 1 } { 113 } \left[ \begin{array} { c c } - 13 & - 7 \\- 5 & - 6\end{array} \right]$$

Question 5

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$$ to find the inverse of the matrix (if it exists). ​​ $$\left[ \begin{array} { c c } - 7 & - 9 \\7 & 9\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { c c } - 9 & - 9 \\7 & 7\end{array} \right]$$
B)​ $$\left[ \begin{array} { r r } - 9 & - 9 \\- 7 & - 7\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } - 9 & 9 \\7 & - 7\end{array} \right]$$
D) $$\left[ \begin{array} { c c } 9 & - 9 \\7 & 7\end{array} \right]$$
E)​Does not exist

Question 6

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists). ​​ $$\left[ \begin{array} { c c } 1 & - 2 \\- 3 & 2\end{array} \right]$$ ​

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

Question 7

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { l l } 3 & 0 \\0 & 4\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { l l } \frac { 1 } { 4 } & 0 \\0 & \frac { 1 } { 4 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l l } \frac { 1 } { 4 } & 0 \\0 & 0\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } 0 & \frac { 1 } { 4 } \\\frac { 1 } { 4 } & 0\end{array} \right]$$
D)​ $$\left[ \begin{array} { l l } \frac { 1 } { 4 } & 0 \\0 & \frac { 1 } { 3 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l l } \frac { 1 } { 3 } & 0 \\0 & \frac { 1 } { 4 }\end{array} \right]$$

Question 8

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { l l } 6 & - 7 \\7 & - 8\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { c c } - 8 & 7 \\7 & 6\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c } - 8 & 7 \\- 7 & 6\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } - 8 & - 7 \\- 7 & 6\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c } - 8 & 7 \\7 & - 6\end{array} \right]$$
E)​ $$\left[ \begin{array} { c c } - 8 & - 7 \\7 & 6\end{array} \right]$$

Question 9

Use the inverse formula $$A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$$ to find the inverse of the 2×2 matrix (if it exists). ​​ $$\left[ \begin{array} { c c } - 12 & 3 \\5 & 2\end{array} \right]$$ ​

A)​ $$- \frac { 1 } { 39 } \left[ \begin{array} { c c } 2 & - 3 \\- 5 & - 12\end{array} \right]$$
B)​ $$- \frac { 1 } { 39 } \left[ \begin{array} { c c } - 2 & - 3 \\5 & 12\end{array} \right]$$
C)​ $$- \frac { 1 } { 39 } \left[ \begin{array} { c c } - 2 & 3 \\5 & - 12\end{array} \right]$$
D)​ $$- \frac { 1 } { 39 } \left[ \begin{array} { c c } 2 & - 3 \\5 & 12\end{array} \right]$$
E)​ $$- \frac { 1 } { 39 } \left[ \begin{array} { l l } - 2 & - 3 \\- 5 & - 12\end{array} \right]$$

Question 10

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c } 4 & - 1 \\- 3 & 1\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { l l } 1 & 1 \\3 & 4\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c } 1 & 1 \\- 3 & 4\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } - 1 & 1 \\3 & 4\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c } 1 & 1 \\3 & - 4\end{array} \right]$$
E)Does not exist

Question 11

Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l } 3 x + 4 y = - 2 \\5 x + 3 y = 4\end{array} \right.$$ ​

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

Question 12

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c c } 4 & 0 & 0 \\6 & 0 & 0 \\5 & 11 & 11\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { c c c } - 4 & 0 & 0 \\- 6 & 0 & 0 \\5 & 4 & 11\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c c } - 4 & 0 & 0 \\- 6 & 0 & 0 \\- 5 & - 4 & - 11\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c c } 4 & 0 & 0 \\6 & 0 & 0 \\5 & 4 & 11\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c c } - 4 & 0 & 0 \\- 6 & 0 & 0 \\- 5 & 4 & 11\end{array} \right]$$
E)Does not exist

Question 13

Solve the system of linear equations.​ $$\left\{ \begin{array} { c } x + y + z = - 7 \\3 x + 5 y + 4 z = 8 \\3 x + 6 y + 5 z = 0\end{array} \right.$$ ​

A)​ $( 2,37 , - 45 )$
B)​ $( - 1,37 , - 45 )$
C)​ $( - 1 , - 37 , - 45 )$
D)​ $( 1,37 , - 45 )$
E)​ $( 1,38 , - 45 )$

Question 14

Solve the system of linear equations.​ $$\left\{ \begin{array} { c } x - 2 y = 6 \\2 x - 3 y = 4\end{array} \right.$$ ​ ​

A)​ $( - 10 , - 8 )$
B)​ $( - 10,8 )$
C)​ $( - 10,4 )$
D)​ $( - 4 , - 10 )$
E)​ $( 10,8 )$

Question 15

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c } - 7 & 33 \\4 & - 19\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { c c } - 19 & - 33 \\4 & - 7\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c } - 19 & - 33 \\- 4 & - 7\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } 19 & - 33 \\- 4 & - 7\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c } 19 & - 33 \\4 & - 7\end{array} \right]$$
E)​ $$\left[ \begin{array} { c c } 19 & - 33 \\4 & 7\end{array} \right]$$

Question 16

Solve the system of linear equations.​ $$\left\{ \begin{array} { l l } x - 2 y & = 5 \\2 x - 3 y & = 10\end{array} \right.$$ ​

A)​ $( 5,10 )$
B)​ $( 5,5 )$
C)​ $( 10,5 )$
D)​ $( 5,0 )$
E)​ $( 0,5 )$

Question 17

Solve the system of linear equations.​ $$\left\{ \begin{array} { l } x - 2 y = 3 \\2 x - 3 y = - 4\end{array} \right.$$ ​

A)​ $( - 17,4 )$
B)​ $( - 17,10 )$
C)​ $( - 4 , - 17 )$
D)​ $( 17,10 )$
E)​ $( - 17 , - 10 )$

Question 18

Find the inverse of the matrix (if it exists).​ $$\left[ \begin{array} { c c c } - 5 & 0 & 0 \\2 & 0 & 0 \\1 & 5 & 7\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { l l l } 5 & 0 & 0 \\2 & 0 & 0 \\1 & 5 & 7\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c c } - 5 & 0 & 0 \\- 2 & 0 & 0 \\1 & 5 & 7\end{array} \right]$$
C) $$\left[ \begin{array} { c c c } - 5 & 0 & 0 \\- 2 & 0 & 0 \\- 1 & - 5 & - 7\end{array} \right]$$
D)​ $$\left[ \begin{array} { l l l } - 5 & 0 & 0 \\- 2 & 0 & 0 \\- 1 & 5 & 7\end{array} \right]$$
E)​Does not exist

Question 19

Solve the system of linear equations.​ $$\left\{ \begin{array} { l } x + y + z = 0 \\3 x + 5 y + 4 z = 9 \\3 x + 6 y + 5 z = 6\end{array} \right.$$ ​

A)​ $( 3,12 , - 15 )$
B)​ $( 3,12,15 )$
C)​ $( - 3 , - 12 , - 15 )$
D)​ $( 3 , - 12 , - 15 )$
E)​ $( - 3,12 , - 15 )$

Question 20

​Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l } 18 x + 12 y = 14 \\30 x + 24 y = 24\end{array} \right.$$

A)​( $\frac { 5 } { 6 }$ , $\frac { 1 } { 6 }$ )
B)​( 4, $\frac { 1 } { 6 }$ )
C)​( $\frac { 2 } { 3 }$ , $\frac { 1 } { 6 }$ )
D)​( $\frac { 2 } { 3 }$ , $\frac { 1 } { 4 }$ )
E)​( $\frac { 2 } { 3 }$ ,3 )

Question 21

The table shows the enrollment projections (in millions)for public universities in the United States for the years 2010 through 2012.
Year
Enrollment projections
2010
13)83
2011
14)07
2012
14)23
The data can be modeled by the quadratic function $y=a t^{2}+b t+c$ .Create a system of linear equations for the data.Let t represent the year,with t = 10 corresponding to 2010.

A) $$\left\{ \begin{array} { l } 100 a + 10 b + c = 13.83 \\121 a - 11 b + c = 14.07 \\144 a + 12 b + c = 14.23\end{array} \right.$$
B)​ $$\left\{ \begin{array} { l } 100 a - 10 b + c = 13.83 \\121 a + 11 b + c = 14.07 \\144 a + 12 b + c = 14.23\end{array} \right.$$
C)​ $$\left\{ \begin{array} { l } 100 a + 10 b + c = 13.83 \\121 a + 11 b + c = 14.07 \\144 a + 12 b + c = 14.23\end{array} \right.$$
D)​ $$\left\{ \begin{array} { l } 100 a + 10 b + c = 13.83 \\121 a + 11 b + c = 14.07 \\144 a + 12 b - c = 14.23\end{array} \right.$$
E)​ $$\left\{ \begin{array} { l } 100 a - 10 b - c = 13.83 \\121 a - 11 b - c = 14.07 \\144 a - 12 b - c = 14.23\end{array} \right.$$

Question 22

Consider a person who invests in AAA-rated bonds,A-rated bonds,and B-rated bonds.The average yields are 6.5% on AAA bonds,7% on A bonds,and 9% on B bonds.The person invests twice as much in B bonds as in A bonds.Let x,y and z represent the amounts invested in AAA,A,and B bonds,respectively. Total Investment
Annual Return
$12,000
890 $$\left\{ \begin{array} { c l } x + y + z & = 12,000 \text { (total investment) } \\0.065 x + 0.07 y + 0.09 z & = 890 \text { (annual retum) } \\2 y - z & = 0\end{array} \right.$$ Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.

A)$7,000 in AAA-rated bonds $2,000 in A-rated bonds
$4,000 in B-rated bonds
B)$5,000 in AAA-rated bonds $7,000 in A-rated bonds
$3,000 in B-rated bonds
C)$5,000 in AAA-rated bonds $6,000 in A-rated bonds
$4,000 in B-rated bonds
D)$3,000 in AAA-rated bonds $2,000 in A-rated bonds
$6,000 in B-rated bonds
E)$6,000 in AAA-rated bonds $2,000 in A-rated bonds
$4,000 in B-rated bonds

Question 23

Find the inverse of the matrix $$\left[ \begin{array} { c c } 2 & 4 \\- 6 & - 10\end{array} \right]$$ .

A)​ $$\frac { 1 } { 13 } \left[ \begin{array} { l l } - 2 & - 4 \\- 6 & 10\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c } - 2 & - 4 \\10 & 6\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } 2 & - 6 \\8 & - 14\end{array} \right]$$
D)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 2 & 4 \\- 6 & - 10\end{array} \right]$$
E)​ $$\frac { 1 } { 2 } \left[ \begin{array} { c c } - 5 & - 2 \\3 & 1\end{array} \right]$$

Question 24

Find the inverse of the matrix $$\left[ \begin{array} { c c } - 1 & 4 \\3 & - 2\end{array} \right]$$ (if it exists).

A)​ $$- \frac { 1 } { - 10 } \left[ \begin{array} { l l } - 2 & - 4 \\- 3 & - 1\end{array} \right]$$
B) $$- \frac { 1 } { - 10 } \left[ \begin{array} { c c } - 1 & 4 \\3 & - 2\end{array} \right]$$
C)​ $$\frac { 1 } { - 10 } \left[ \begin{array} { l l } - 2 & - 4 \\- 3 & - 1\end{array} \right]$$
D)​ $$\frac { 1 } { - 10 } \left[ \begin{array} { l l } 2 & 4 \\3 & 1\end{array} \right]$$
E)​does not exist

Question 25

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​
875 units of beef
830 units of chicken
850 units of liver
​

A)765 muffins,5 bones,50 cookies
B)50 muffins,765 bones,50 cookies
C)5 muffins,765 bones,50 cookies
D)50 muffins,765 bones,5 cookies
E)5 muffins,5 bones,830 cookies

Question 26

A florist is creating 10 centerpieces for the tables at a wedding reception.Roses cost $2.50 each,lilies cost $8 each,and irises cost $4 each.The customer has a budget of $300 allocated for the centerpieces and wants each centerpiece to contain 12 flowers,with twice as many roses as the number of irises and lilies combined. ​
Write a system of linear equations that represents the situation.
​

A) $$\left\{ \begin{array} { l l } 2.5 r - 8 l - 4 i & = 300 \\- r - 2 l - 2 i & = 0 \\r - l - i & = 120\end{array} \right.$$
B)​ $$\left\{ \begin{array} { l l } 2.5 r + 8 l + 4 i & = 300 \\r + 2 l + 2 i & = 0 \\r + l + i & = 120\end{array} \right.$$
C)​ $$\left\{ \begin{array} { l l } 2.5 r + 8 l + 4 i & = 300 \\- r + 2 l + 2 i & = 0 \\r + l + i & = 120\end{array} \right.$$
D)​ $$\left\{ \begin{array} { l l } 2.5 r - 8 l + 4 i & = 300 \\- r + 2 l + 2 i & = 0 \\r + l + i & = 120\end{array} \right.$$
E)​ $$\left\{ \begin{array} { l l } 2.5 r + 8 l - 4 i & = 300 \\- r + 2 l + 2 i & = 0 \\r + l + i & = 120\end{array} \right.$$

Question 27

​Solve the system of linear equations $$\left\{ \begin{array} { l } 2 x + 3 y = - 15 \\4 x - 3 y = 10\end{array} \right.$$ using the inverse matrix $$\left[ \begin{array} { c c } \frac { 1 } { 6 } & \frac { 1 } { 6 } \\\\\frac { 2 } { 9 } & - \frac { 1 } { 9 }\end{array} \right]$$

A)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } \frac { 5 } { 6 } \\\\\frac { 5 } { 2 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } - \frac { 5 } { 6 } \\\\- \frac { 40 } { 9 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { 25 } { 18 } \\\\\frac { 5 } { 2 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } \frac { 5 } { 2 } \\0\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } - \frac { 5 } { 3 } \\\\\frac { 5 } { 6 }\end{array} \right]$$

Question 28

Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l } 0.2 x - 0.6 y = 14.4 \\- x + 1.4 y = - 20.8\end{array} \right.$$ ​

A)(-24,-31)
B)(-23,-32)
C)(-22,-32)
D)(-24,-32)
E)(-23,-30)

Question 29

​Use an inverse matrix to solve (if possible)the system of linear equations.​ $$\left\{ \begin{array} { l l } 1.8 x - 5 y & = 4 \\28.8 x - 16 y & = 5\end{array} \right.$$ ​

A)​( $\frac { 1 } { 16 }$ ,16)
B)​(0,0)
C)​( $\frac { 1 } { 16 }$ , $\frac { 5 } { 16 }$ )
D)​(16,16)
E)​No solution

Question 30

A coffee manufacturer sells a 14-pound package that contains three flavors of coffee for $25.French vanilla coffee costs $5 per pound,hazelnut flavored coffee costs $5.50 per pound,and Swiss chocolate flavored coffee costs $6 per pound.The package contains the same amount of hazelnut as Swiss chocolate.Let f represent the number of pounds of French vanilla,h represent the number of pounds of hazelnut,and s represent the number of pounds of Swiss chocolate. ​
Write a system of linear equations that represents the situation.
​

A) $$\left\{ \begin{array} { c l } 5 f - 5.5 h + 6 s & = 25 \\f + h + s & = 14 \\h - s & = 0\end{array} \right.$$ ​
B)​ $$\left\{ \begin{aligned}5 f + 5.5 h + 6 s & = 25 \\f + h + s & = 14 \\h - s & = 0\end{aligned} \right.$$
C)​ $$\left\{ \begin{array} { c l } 5 f - 5.5 h + 6 s & = 25 \\f - h + s & = 14 \\h - s & = 0\end{array} \right.$$
D)​ $$\left\{ \begin{array} { c l } 5 f - 5.5 h + 6 s & = 25 \\f + h + s & = 14 \\h + s & = 0\end{array} \right.$$
E)​ $$\left\{ \begin{array} { c l } 5 f - 5.5 h - 6 s & = 25 \\f - h - s & = 14 \\h + s & = 0\end{array} \right.$$

Question 31

Find the inverse of the matrix $$\left[ \begin{array} { c c } - 16 & 22 \\- 8 & 11\end{array} \right]$$ (if it exists).

A)does not exist
B)​ $- \frac { 1 } { 16 }$ $\left[ \begin{array} { c c } 11 & - 22 \\8 & - 16\end{array} \right]$
C)​ $\frac { 1 } { 11 }$ $\left[ \begin{array} { c c } - 16 & 22 \\- 8 & 11\end{array} \right]$
D)​ $- \frac { 1 } { 22 }$ $\left[ \begin{array} { c c } - 11 & 22 \\- 8 & 16\end{array} \right]$
E)​ $\frac { 1 } { 8 }$ $\left[ \begin{array} { c c } 11 & - 22 \\8 & - 16\end{array} \right]$

Question 32

Consider a person who invests in AAA-rated bonds,A-rated bonds,and B-rated bonds.The average yields are 6.5% on AAA bonds,7% on A bonds,and 9% on B bonds.The person invests twice as much in B bonds as in A bonds.Let x,y and z represent the amounts invested in AAA,A,and B bonds,respectively. Total Investment
Annual Return
$32,000
2465 $$\left\{ \begin{array} { c l } x + y + z & = 32,000 \text { (total investment) } \\0.065 x + 0.07 y + 0.09 z & = 2465 \text { (annual retur) } \\2 y - z & = 0\end{array} \right.$$ Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.
​

A)$11,000 in AAA-rated bonds $7,000 in A-rated bonds
$14,000 in B-rated bonds
B)$11,000 in AAA-rated bonds $14,000 in A-rated bonds
$7,000 in B-rated bonds
C)$14,000 in AAA-rated bonds $11,000 in A-rated bonds
$7,000 in B-rated bonds
D)$14,000 in AAA-rated bonds $7,000 in A-rated bonds
$11,000 in B-rated bonds
E)$7,000 in AAA-rated bonds $11,000 in A-rated bonds
$14,000 in B-rated bonds

Question 33

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​
900 units of beef
700 units of chicken
800 units of liver
​

A)200 muffins,500 bones,200 cookies
B)200 muffins,500 bones,0 cookies
C)0 muffins,0 bones,700 cookies
D)500 muffins,0 bones,200 cookies
E)0 muffins,500 bones,200 cookies

Question 34

​Solve the system of linear equations $$\left\{ \begin{array} { l } 3 x + 6 y = 1 \\6 x + 8 y = 2\end{array} \right.$$ using the inverse matrix $$\left[ \begin{array} { c c } - \frac { 2 } { 3 } & \frac { 1 } { 2 } \\\frac { 1 } { 2 } & - \frac { 1 } { 4 }\end{array} \right]$$

A)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { 11 } { 12 } \\0\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } \frac { 1 } { 3 } \\0\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { 11 } { 12 } \\\\\frac { 1 } { 3 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { 1 } { 3 } \\- \frac { 1 } { 2 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } - \frac { 1 } { 2 } \\\frac { 1 } { 3 }\end{array} \right]$$

Question 35

Consider a person who invests in AAA-rated bonds,A-rated bonds,and B-rated bonds.The average yields are 6.5% on AAA bonds,7% on A bonds,and 9% on B bonds.The person invests twice as much in B bonds as in A bonds.Let x,y and z represent the amounts invested in AAA,A,and B bonds,respectively. $$\begin{array} { | c | c | } \hline \text { Total Investment } & \text { Annual Return } \\\hline \$ 12,000 & 890 \\\hline\end{array}$$ $$\left\{ \begin{array} { c l } x + y + z & = 663,000 \text { (total investment) } \\0.065 x + 0.07 y + 0.09 z & = 47,000 \text { (annual retum) } \\2 y - z & = 0\end{array} \right.$$ Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.
​

A)$142,000 in AAA-rated bonds $451,000 in A-rated bonds
$71,000 in B-rated bonds
B)$450,000 in AAA-rated bonds $71,000 in A-rated bonds
$142,000 in B-rated bonds
C)$143,000 in AAA-rated bonds $71,000 in A-rated bonds
$450,000 in B-rated bonds
D)$72,000 in AAA-rated bonds $450,000 in A-rated bonds
$142,000 in B-rated bonds
E)$450,000 in AAA-rated bonds $142,000 in A-rated bonds
$70,000 in B-rated bonds

Question 36

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $$\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]$$ (if it exists).

A)​ $$\left[ \begin{array} { c c c } - 9 & 12 & 6 \\0 & - 3 & 0 \\0 & 3 & - 9\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c c } - 6 & 6 & 9 \\3 & - 3 & 0 \\0 & 3 & 12\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c c } - 6 & 0 & 9 \\0 & - 3 & 0 \\0 & 3 & - 6\end{array} \right]$$
D) $$\left[ \begin{array} { c c c } 3 & 3 & 9 \\0 & - 3 & 0 \\0 & 6 & 12\end{array} \right]$$
E)​does not exist ​

Question 37

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​
3,000 units of beef
2,950 units of chicken
2,900 units of liver
​

A)200 muffins,2,300 bones,200 cookies
B)150 muffins,150 bones,2,950 cookies
C)200 muffins,2,300 bones,150 cookies
D)2,300 muffins,150 bones,200 cookies
E)150 muffins,2,300 bones,200 cookies

Question 38

Find the inverse of the matrix $$\left[ \begin{array} { c c c } 7 & 7 & 7 \\21 & 35 & 28 \\21 & 42 & 35\end{array} \right]$$ . ​

A)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 1 & 1 & - 1 \\- 3 & 2 & - 1 \\3 & - 3 & 2\end{array} \right]$$
B)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 1 & 1 & 1 \\0 & - 2 & - 1 \\3 & - 3 & 2\end{array} \right]$$
C)​ $$- 7 \left[ \begin{array} { c c c } 3 & 1 & 0 \\- 3 & 1 & - 1 \\1 & - 3 & 2\end{array} \right]$$
D)​ $$- \frac { 1 } { 7 } \left[ \begin{array} { c c c } 1 & 0 & - 1 \\- 3 & 2 & - 1 \\2 & - 3 & 3\end{array} \right]$$
E)​ $$7 \left[ \begin{array} { c c c } 1 & 1 & 1 \\- 3 & 0 & - 1 \\3 & - 3 & 2\end{array} \right]$$

Question 39

Find the inverse of the following matrix.​ $$\left[ \begin{array} { c c } \cos \theta & \sin \theta \\- \sin \theta & \cos \theta\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { c c } \cos \theta & - \sin \theta \\\sin \theta & \cos \theta\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c } - \cos \theta & - \sin \theta \\\sin \theta & \cos \theta\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } \cos \theta & \sin \theta \\- \sin \theta & \cos \theta\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c } \cos \theta & \sin \theta \\- \sin \theta & - \cos \theta\end{array} \right]$$
E)​ $$\left[ \begin{array} { l l } \cos \theta & - \sin \theta \\\sin \theta & - \cos \theta\end{array} \right]$$

Question 40

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​
800 units of beef
750 units of chicken
725 units of liver
​

A)150 muffins,300 bones,150 cookies
B)150 muffins,300 bones,100 cookies
C)300 muffins,100 bones,150 cookies
D)100 muffins,300 bones,150 cookies
E)100 muffins,100 bones,750 cookies

Question 41

​Solve the system of linear equations $$\left\{ \begin{array} { l l } - 6 x + 18 y + 6 z & = 1 \\12 x + 30 y & = 2 \\18 x + 6 y - 12 z & = - 1\end{array} \right.$$ using an inverse matrix. ​

A)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 1 } { 6 } \\\\\frac { - 1 } { 2 } \\\\\frac { 1 } { 3 }\end{array} \right]$$ ​
B)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 1 } { 6 } \\\\\frac { - 1 } { 2 } \\\\\frac { - 1 } { 3 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 1 } { 6 } \\0 \\\frac { 1 } { 2 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } 0 \\\frac { 1 } { 3 } \\\frac { - 1 } { 6 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { l } \frac { 1 } { 6 } \\0 \\\frac { 1 } { 3 }\end{array} \right]$$

Question 42

​Solve the system of linear equations​ $$\left\{ \begin{array} { l l } 7 x + 7 y + 7 z & = 0 \\21 x + 35 y + 28 z & = 6 \\21 x + 42 y + 35 z & = 1\end{array} \right.$$ ​using the inverse matrix $$\frac { 1 } { 7 } \left[ \begin{array} { c c c } 1 & 1 & - 1 \\- 3 & 2 & - 1 \\3 & - 3 & 2\end{array} \right]$$ .

A)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 5 } { 7 } \\\\\frac { 11 } { 7 } \\\\\frac { - 12 } { 7 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 5 } { 7 } \\\\\frac { 11 } { 7 } \\\\\frac { - 16 } { 7 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 16 } { 7 } \\\\\frac { 5 } { 7 } \\\\\frac { 11 } { 7 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 12 } { 7 } \\\\\frac { 11 } { 7 } \\\\\frac { - 5 } { 7 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 5 } { 7 } \\\\\frac { - 16 } { 7 } \\\\\frac { 11 } { 7 }\end{array} \right]$$

Question 43

Find the inverse of A.​ $$A = \left[ \begin{array} { c c } 2 & 3 \\- 4 & 1\end{array} \right]$$ ​

A)​ $$A ^ { - 1 } = \left[ \begin{array} { c c } \frac { 1 } { 7 } & - \frac { 3 } { 7 } \\\\\frac { 2 } { 7 } & \frac { 1 } { 7 }\end{array} \right]$$
B)​ $$A ^ { - 1 } = \left[ \begin{array} { c c } \frac { 1 } { 14 } & - \frac { 3 } { 14 } \\\\\frac { 2 } { 7 } & \frac { 1 } { 7 }\end{array} \right]$$
C)​ $$A ^ { - 1 } = \left[ \begin{array} { c c } \frac { 1 } { 14 } & - \frac { 3 } { 7 } \\\\\frac { 2 } { 7 } & \frac { 1 } { 14 }\end{array} \right]$$
D)​ $$A ^ { - 1 } = \left[ \begin{array} { c c } \frac { 1 } { 14 } & \frac { 3 } { 14 } \\\frac { 2 } { 7 } & \frac { 1 } { 7 }\end{array} \right]$$
E)​ $$A ^ { - 1 } = \left[ \begin{array} { c c } \frac { 1 } { 7 } & - \frac { 3 } { 7 } \\\frac { 2 } { 14 } & \frac { 1 } { 14 }\end{array} \right]$$

Question 44

Find the inverse of the matrix.​ $$\left[ \begin{array} { c c c } 1 & 0 & 5 \\1 & 1 & 5 \\- 5 & 1 & 1\end{array} \right]$$ ​

A)​​ $$\left[ \begin{array} { c c c } - 29 & 5 & - 5 \\- 1 & 1 & 0 \\6 & 1 & 1\end{array} \right]$$
B) $$\left[ \begin{array} { c c c } 4 & \frac { 5 } { 26 } & - \frac { 5 } { 26 } \\1 & 1 & 0 \\\frac { 6 } { 26 } & - \frac { 1 } { 26 } & \frac { 1 } { 26 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c c } - \frac { 4 } { 26 } & \frac { 5 } { 26 } & - \frac { 5 } { 26 } \\- 1 & 1 & 0 \\\frac { 6 } { 26 } & - \frac { 1 } { 26 } & \frac { 1 } { 26 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c c } 29 & 5 & - 5 \\- 1 & 1 & - 1 \\6 & - 1 & 1\end{array} \right]$$
E)​ $$\left[ \begin{array} { c c c } 29 & 5 & - 5 \\- 1 & 1 & 1 \\6 & - 1 & 1\end{array} \right]$$

Question 45

Find the inverse of the matrix $$\left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right]$$ (if it exists).

A)​ $$\frac { 1 } { 4 } \left[ \begin{array} { l l } 7 & - 14 \\4 & - 8\end{array} \right]$$
B)​ $$- \frac { 1 } { 8 } \left[ \begin{array} { c c } 7 & - 14 \\4 & - 8\end{array} \right]$$
C)​ $$- \frac { 1 } { 4 } \left[ \begin{array} { c c } - 7 & - 14 \\4 & 8\end{array} \right]$$
D)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } - 8 & 14 \\- 4 & 7\end{array} \right]$$
E)does not exist

Question 46

Use a graphing calculator to find the inverse of the matrix.​ $$\left[ \begin{array} { l l l l } 1 & 4 & 8 & 3 \\0 & 1 & 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]$$ ​ ​

A)​ $$\left[ \begin{array} { c c c c } 1 & 4 & 8 & - 3 \\0 & 1 & - 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c c c } 1 & - 4 & 8 & 3 \\0 & 1 & - 4 & 8 \\0 & 0 & 1 & - 4 \\0 & 0 & 0 & 1\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c c c } 1 & - 4 & - 8 & - 3 \\0 & 1 & - 4 & - 8 \\0 & 0 & 1 & - 4 \\0 & 0 & 0 & 1\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c c c } 1 & - 4 & 8 & - 3 \\0 & 1 & - 4 & 8 \\0 & 0 & 1 & - 4 \\0 & 0 & 0 & 1\end{array} \right]$$
E)​ $$\left[ \begin{array} { c c c c } 1 & 4 & 8 & - 3 \\0 & 1 & 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]$$

Question 47

​Solve the system of linear equations $$\left\{ \begin{array} { l } 4 x + 4 y = 3 \\8 x - 20 y = - 9\end{array} \right.$$ using an inverse matrix.

A)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } \frac { 3 } { 14 } \\\\\frac { 15 } { 28 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } \frac { 9 } { 28 } \\\\\frac { 12 } { 7 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { 9 } { 28 } \\\\\frac { - 3 } { 14 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { 9 } { 7 } \\\\\frac { 15 } { 28 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { c } \frac { - 9 } { 7 } \\\\\frac { - 3 } { 14 }\end{array} \right]$$

Question 48

Find the inverse of the matrix.​ $$\left[ \begin{array} { l l l } 1 & 5 & 2 \\0 & 1 & 5 \\0 & 0 & 1\end{array} \right]$$ ​

A)​ $$\left[ \begin{array} { c c c } 1 & 5 & - 23 \\0 & 1 & 5 \\0 & 0 & 1\end{array} \right]$$
B)​ $$\left[ \begin{array} { c c c } - 1 & - 5 & 23 \\0 & - 1 & - 5 \\0 & 0 & - 1\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c c } 1 & - 5 & - 23 \\0 & 1 & - 5 \\0 & 0 & 1\end{array} \right]$$
D)​ $$\left[ \begin{array} { l l l } 1 & 5 & 23 \\0 & 1 & 5 \\0 & 0 & 1\end{array} \right]$$
E)​ $$\left[ \begin{array} { c c c } 1 & - 5 & 23 \\0 & 1 & - 5 \\0 & 0 & 1\end{array} \right]$$

Question 49

Find the inverse of A.​ $$A = \left[ \begin{array} { c c c } - 4 & 12 & 4 \\8 & 20 & 0 \\12 & 4 & - 8\end{array} \right]$$ ​

A)​ $$A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } 10 & 7 & 5 \\4 & - 1 & 2 \\- 13 & 10 & - 11\end{array} \right]$$
B)​ $$A ^ { - 1 } = \frac { 1 } { 24 } \left[ \begin{array} { c c c } 10 & 7 & 5 \\4 & 1 & 2 \\13 & 10 & 11\end{array} \right]$$
C)​ $$A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } - 10 & 7 & - 5 \\4 & - 1 & 2 \\- 13 & 10 & - 11\end{array} \right]$$
D)​ $$A ^ { - 1 } = \frac { 1 } { 36 } \left[ \begin{array} { c c c } - 10 & 7 & - 5 \\4 & - 1 & 2 \\- 13 & 10 & - 11\end{array} \right]$$
E)​ $$A ^ { - 1 } = \frac { 1 } { 28 } \left[ \begin{array} { c c c } 10 & 7 & 5 \\4 & 1 & 2 \\13 & 10 & 11\end{array} \right]$$

Question 50

Find the inverse of the matrix.​ $$\left[ \begin{array} { l l } 4 & 1 \\1 & 4\end{array} \right]$$ ​

A)​​ $$\left[ \begin{array} { c c } \frac { 4 } { 15 } & \frac { 1 } { 15 } \\\\\frac { 1 } { 15 } & \frac { 4 } { 15 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } - \frac { 4 } { 15 } - \frac { 1 } { 15 } \\- \frac { 1 } { 15 } - \frac { 4 } { 15 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } \frac { 4 } { 15 } & - \frac { 1 } { 15 } \\- \frac { 1 } { 15 } & \frac { 4 } { 15 }\end{array} \right.$$
D)​ $$\left[ \begin{array} { c c } - \frac { 4 } { 15 } & \frac { 1 } { 15 } \\\frac { 1 } { 15 } & - \frac { 4 } { 15 }\end{array} \right]$$ ​
E)​ $$\left[ \begin{array} { c c } \frac { 4 } { 15 } & 0 \\0 & \frac { 4 } { 15 }\end{array} \right]$$

Question 51

​Solve the system of linear equations $$\left\{ \begin{array} { l l } 4 x _ { 1 } - 8 x _ { 2 } - 4 x _ { 3 } - 8 x _ { 4 } & = 0 \\12 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 12 x _ { 4 } & = - 15 \\8 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 20 x _ { 4 } & = 10 \\- 4 x _ { 1 } + 16 x _ { 2 } + 16 x _ { 3 } + 44 x _ { 4 } & = 0\end{array} \right.$$ using the inverse matrix $$\frac { 1 } { 4 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\- 10 & 3 & 0 & - 1 \\- 29 & 7 & 3 & - 2 \\12 & - 3 & - 1 & 1\end{array} \right]$$ .

A)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { - 95 } { 4 } \\\\\frac { - 45 } { 4 } \\\\\frac { - 75 } { 4 } \\\\\frac { 35 } { 4 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } 0 \\\frac { - 5 } { 4 } \\0 \\\frac { - 5 } { 2 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { - 45 } { 4 } \\\\\frac { - 95 } { 4 } \\\\0 \\\\\frac { 35 } { 4 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { r } \frac { - 15 } { 4 } \\\frac { 5 } { 4 } \\0 \\\frac { 5 } { 2 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { - 35 } { 4 } \\\\\frac { - 25 } { 4 } \\\\\frac { 45 } { 4 } \\\\\frac { - 45 } { 4 }\end{array} \right]$$

Question 52

​Solve the system of linear equations $$\left\{ \begin{array} { l l } 5 x _ { 1 } - 10 x _ { 2 } - 5 x _ { 3 } - 10 x _ { 4 } & = 0 \\15 x _ { 1 } - 25 x _ { 2 } - 10 x _ { 3 } - 15 x _ { 4 } & = 8 \\10 x _ { 1 } - 25 x _ { 2 } - 10 x _ { 3 } - 25 x _ { 4 } & = - 8 \\- 5 x _ { 1 } + 20 x _ { 2 } + 20 x _ { 3 } + 55 x _ { 4 } & = 16\end{array} \right.$$ using the inverse matrix $$\frac { 1 } { 5 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\- 10 & 3 & 0 & - 1 \\- 29 & 7 & 3 & - 2 \\12 & - 3 & - 1 & 1\end{array} \right]$$ .

A)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { 16 } { 5 } \\\\\frac { 8 } { 5 } \\\\\frac { - 8 } { 5 } \\\\\frac { - 16 } { 5 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } 0 \\\\\frac { 8 } { 5 } \\\\0 \\\\\frac { 16 } { 5 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { 16 } { 5 } \\\\\frac { 8 } { 5 } \\\\0 \\\\0\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { 24 } { 5 } \\\\\frac { - 8 } { 5 } \\\\0 \\\\\frac { - 16 } { 5 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 } \\x _ { 4 }\end{array} \right] = \left[ \begin{array} { c } \frac { 24 } { 5 } \\\\\frac { - 8 } { 5 } \\\\\frac { - 16 } { 5 } \\\\\frac { 16 } { 5 }\end{array} \right]$$

Question 53

Find the inverse of the matrix $$\left[ \begin{array} { c c } 3 & 6 \\- 9 & - 15\end{array} \right]$$ .

A)​ $$\frac { 1 } { 7 } \left[ \begin{array} { c c } 3 & 6 \\- 9 & - 15\end{array} \right]$$
B)​ $$\frac { 1 } { 3 } \left[ \begin{array} { c c } - 5 & - 2 \\3 & 1\end{array} \right]$$
C)​ $$\left[ \begin{array} { c c } - 3 & - 6 \\15 & 9\end{array} \right]$$
D)​ $$\left[ \begin{array} { c c } 3 & - 9 \\12 & - 21\end{array} \right]$$
E)​ $$\frac { 1 } { 13 } \left[ \begin{array} { l l } - 3 & - 6 \\- 9 & 15\end{array} \right]$$

Question 54

Find the inverse of the matrix $$\left[ \begin{array} { c c c } 8 & 8 & 8 \\24 & 40 & 32 \\24 & 48 & 40\end{array} \right]$$ .

A)​ $$\frac { 1 } { 8 } \left[ \begin{array} { c c c } 1 & 1 & - 1 \\- 3 & 2 & - 1 \\3 & - 3 & 2\end{array} \right]$$
B)​ $$\frac { 1 } { 8 } \left[ \begin{array} { c c c } 1 & 1 & 1 \\0 & - 2 & - 1 \\3 & - 3 & 2\end{array} \right]$$
C)​ $$8 \left[ \begin{array} { c c c } 1 & 1 & 1 \\- 3 & 0 & - 1 \\3 & - 3 & 2\end{array} \right]$$
D)​ $$- \frac { 1 } { 8 } \left[ \begin{array} { c c c } 1 & 0 & - 1 \\- 3 & 2 & - 1 \\2 & - 3 & 3\end{array} \right]$$
E)​ $$- 8 \left[ \begin{array} { c c c } 3 & 1 & 0 \\- 3 & 1 & - 1 \\1 & - 3 & 2\end{array} \right]$$

Question 55

​Use the matrix capabilities of a graphing utility to solve the following system of linear equations: $$\left\{ \begin{array} { c c c } 15 x - 5 y & = & 3 \\10 x + 10 y & = & 6 \\20 z & = & - 12\end{array} \right.$$

A)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 3 } { 10 } \\\\\frac { 9 } { 10 } \\\\\frac { - 3 } { 5 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 9 } { 10 } \\0 \\\frac { - 3 } { 5 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 3 } { 10 } \\\\\frac { 3 } { 10 } \\\\\frac { - 3 } { 5 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 3 } { 10 } \\\\\frac { - 3 } { 10 } \\\\0\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 3 } { 10 } \\\\\frac { 3 } { 10 } \\\\\frac { - 3 } { 5 }\end{array} \right]$$

Question 56

​Use the matrix capabilities of a graphing utility to solve the following system of linear equations:​ $$\left\{ \begin{array} { l l } - 14 x + 14 y + 21 z & = 10 \\7 x - 7 y & = - 5 \\7 x + 28 z & = 20\end{array} \right.$$ ​

A)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 20 } { 7 } \\\frac { 25 } { 7 } \\0\end{array} \right]$$ ​
B)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 10 } { 7 } \\0 \\\frac { - 15 } { 7 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 15 } { 7 } \\\frac { 10 } { 7 } \\\frac { 5 } { 7 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 5 } { 7 } \\\\\frac { - 20 } { 7 } \\\\\frac { 10 } { 7 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 15 } { 7 } \\\\\frac { - 10 } { 7 } \\\\ \frac { 10 } { 7 }\end{array} \right]$$

Question 57

Find the inverse of the matrix $$\left[ \begin{array} { c c } - 2 & 1 \\5 & 4\end{array} \right]$$ (if it exists).

A)​ $$\frac { 1 } { 13 } \left[ \begin{array} { c c } 4 & - 1 \\- 5 & - 2\end{array} \right]$$
B)​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } 4 & - 1 \\- 5 & - 2\end{array} \right]$$
C)​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } - 4 & 1 \\5 & 2\end{array} \right]$$
D)​ $$- \frac { 1 } { 13 } \left[ \begin{array} { c c } - 2 & 1 \\5 & 4\end{array} \right]$$
E)does not exist

Question 58

Show that B is the inverse of A.Show all your work. ​

Question 59

​Solve the system of linear equations​ $$\left\{ \begin{array} { l l } 9 x + 9 y + 9 z & = 1 \\27 x + 45 y + 36 z & = - 3 \\27 x + 54 y + 45 z & = 2\end{array} \right.$$ ​using the inverse matrix $$\frac { 1 } { 9 } \left[ \begin{array} { c c c } 1 & 1 & - 1 \\- 3 & 2 & - 1 \\3 & - 3 & 2\end{array} \right]$$ .

A)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 5 } { 9 } \\\\\frac { - 2 } { 3 } \\\\\frac { 4 } { 9 }\end{array} \right]$$
B)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 4 } { 9 } \\\\\frac { - 11 } { 9 } \\\\\frac { 16 } { 9 }\end{array} \right]$$
C)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { - 11 } { 9 } \\\\\\frac { 16 } { 9 } \\\\\frac { - 4 } { 9 }\end{array} \right]$$
D)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 5 } { 9 } \\\\\frac { 1 } { 3 } \\\\\frac { - 2 } { 9 }\end{array} \right]$$
E)​ $$\left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } \frac { 16 } { 9 } \\\\\frac { - 4 } { 9 } \\\\\frac { - 11 } { 9 }\end{array} \right]$$