Deck 11: Matrices and Determinants

Question 1

Solve $\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right]$ for $x$ and $y$ .

Answer 1

$\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { c c } 2 y - 2 x & 6 + 2 x \\6 y & 18\end{array} \right]$ $\Rightarrow$

$\left[ \begin{array} { c c } 2 y - 2 x & 6 + 2 x \\6 y & 18\end{array} \right] = \left[ \begin{array} { c c } 1 & 3 \\- 6 & 18\end{array} \right]$ $\Rightarrow$

$\left\{ \begin{array} { l } 2 y - 2 x = 1 \\6 + 2 x = 3\end{array} \right.$ $\Rightarrow$ $x = - \frac { 3 } { 2 }$ , $y = - 1$

Question 2

Find the complete solution of the system.

Answer 2

,
,

Question 3

Find the complete solution of the system.

Answer 3

,
,

Question 4

Find the determinant of the matrix
, if it exists.

Question 5

Find the determinant and, if possible, the inverse of the matrix.

Question 6

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.$

Question 7

A man invests a total of $\$ 21,000$ in two accounts, one paying $5.5 \%$ simple interest per year and the other paying $8.5 \%$ simple interest per year. His annual interest from both accounts is $\$ 1,280$ . How much did he invest at each rate?

Question 8

Find the complete solution of the system.

Question 9

Find the dimension of matrix A.

Question 10

Write the system of equations for which the given matrix is the augmented matrix.

Question 11

Solve the system.

Question 12

Given $A = \left[ \begin{array} { c c } 3 & - 1 \\2 & 2\end{array} \right]$
and $B = \left[ \begin{array} { c c } 2 & 3 \\- 1 & 5\end{array} \right]$
, find $A B$ , or explain why the operation cannot be performed.

Question 13

Solve the system of equations $\left\{ \begin{array} { r } 3 x + 2 y = 8 \\x - 2 y = 0\end{array} \right.$
by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c } \frac { 1 } { 4 } & \frac { 1 } { 4 } \\\frac { 1 } { 3 } & - \frac { 3 } { 3 }\end{array} \right]$ .

Question 14

Determine whether the system is linear or nonlinear.

Question 15

Solve the system. $\left\{ \begin{array} { l } x + y = 6 \\x - 2 y = 0\end{array} \right.$

Question 16

Write the system of equations in matrix form.

Question 17

Given $A = \left[ \begin{array} { c c } - 9 & 8 \\- 6 & - 5 \\10 & - 16\end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \\2 & 3 \\0 & 4\end{array} \right]$

, solve $2 X - A = 3 B$ , for $X$ , or explain why there is no solution.

Question 18

Given $A = \left[ \begin{array} { l l } 0 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 1 & 5 \\- 2 & 3\end{array} \right]$ , $C = \left[ \begin{array} { l l l } 0 & 1 & 2 \\6 & 2 & 3\end{array} \right]$ , $D = \left[ \begin{array} { l } 1 \\5 \\0\end{array} \right]$ , find $( A B ) \cdot ( C D )$ , or explain why the operation cannot be performed.

Question 19

Find the quadratic polynomial $y = ax ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$ , $( - 1 , - 6 )$ , and $( 3 , - 6 )$ .

Question 20

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither.

Question 21

Write out the form of the partial fraction decomposition.

Question 22

Solve for
and
.

Question 23

Given $A = \left[ \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 4 \\3 & 8\end{array} \right]$

, solve $X - 2 A = B$ , for the matrix $X$ , or explain why there is no solution.

Question 24

Find the dimension of matrix A.

Question 25

Find the partial fraction decomposition of
.

Question 26

Find the quadratic polynomial $y = c x ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$ , $( - 1 , - 6 )$ , and $( 3 , - 6 )$ .

Question 27

Given
,
,
,
, find
, or explain why the operation cannot be performed.

Question 28

Solve the system.

Question 29

Determine whether the system is linear or nonlinear.

Question 30

Graph the inequality
.

Question 31

Find the complete solution of the system.

Question 32

Graph the solution of the system of inequalities.

Question 33

Find the complete solution of the system.

Question 34

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither.

Question 35

Sketch the triangle with vertices
,
, and
, and use a determinant to find its area.

Question 36

Solve the system.

Question 37

Given $A = \left[ \begin{array} { c c } 1 & 5 \\- 2 & 3\end{array} \right]$
and $B = \left[ \begin{array} { c c c } 4 & 1 & - 1 \\3 & - 5 & 2\end{array} \right]$ , find $B A$ , or explain why the operation cannot be performed.

Question 38

Find the complete solution of the system.

Question 39

Gary invests a total of $\$ 21,000$ in two accounts, one paying $5.5 \%$ simple interest per year and the other paying $8.5 \%$ simple interest per year. His annual interest from both accounts is $\$ 1,280$ . How much did he invest at each rate?

Question 40

Write the system of equations for which the given matrix is the augmented matrix.

Question 41

Find the complete solution of the system.

A)
B)
C)
D)
E)none

Question 42

Write the system of equations in matrix form.

Question 43

Solve the system.

A)
B)
C)
D)
E)none

Question 44

Find the determinant of the matrix
, if it exists

Question 45

Write out the form of the partial fraction decomposition of
.

Question 46

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither.

A)row-echelon form
B)reduced row-echelon form
C)neither

Question 47

Find the determinant and, if possible, the inverse of the matrix.

Question 48

Find the dimension of matrix A.

A)
B)
C)
D)
E)none

Question 49

Find the partial fraction decomposition of
.

Question 50

Graph the solution of the system of inequalities.

Question 51

Find the complete solution of the system.

A)
B)
C)
D)
E)none

Question 52

The system is linear. $$\left\{ \begin{array} { r } x + \sqrt { 2 } y + z + w = e ^ { 2 } \\3 x + 2 y - z - w = 0 \\x + e ^ { 2 } y + z + 2 w = 1 \\x + y + z + w = 1\end{array} \right.$$

Question 53

Solve the system of equations
by converting to a matrix equation and using its inverse coefficient matrix
.

Question 54

Solve the system.

A)
B)
C)
D)
E)

Question 55

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + y - 3 z = 0 \\x + y + z = 4 \\x - y = 2\end{array} \right.$

Question 56

Write the system of equations for which the given matrix is the augmented matrix. $$\left[ \begin{array} { c c c c } 0 & 2 & 6 & 3 \\1 & 0 & 0 & - 9 \\2 & 3 & 0 & 0\end{array} \right]$$

A) $\left\{ \begin{aligned}y + 6 z & = 3 \\z & = - 9 \\2 x - 3 y & = 0\end{aligned} \right.$

B) $\left\{ \begin{aligned}2 x + 6 z & = 3 \\x & = - 9 \\x + 3 y & = 0\end{aligned} \right.$

C) $\left\{ \begin{aligned}2 x + 6 z & = 3 \\z & = - 9 \\2 y + 3 z & = 0\end{aligned} \right.$

D) $\left\{ \begin{aligned}2 y + 6 z & = 3 \\x & = - 9 \\2 x + 3 y & = 0\end{aligned} \right.$

E)none

Question 57

A boat travels upstream (against the current) between two points on a river mi apart in
Hours. The return trip downstream takes only
Minutes. What is the boat's speed, and how fast does the current in the river flow?

A)The speed of the boat is
Mi/h and the speed of the current flow is about
Mi/h
B)The speed of the boat is
Mi/h and the speed of the current flow is about
Mi/h
C)The speed of the boat is
Mi/h and the speed of the current flow is about
Mi/h
D)
C)The speed of the boat is
Mi/h and the speed of the current flow is about
Mi/h
E)The speed of the boat is
Mi/h and the speed of the current flow is about
Mi/h.

Question 58

Find the complete solution of the system. $$\left\{ \begin{array} { l } x + z = 4 \\y + 2 z = 1 \\- y + z = 0\end{array} \right.$$

A) $x = \frac { 11 } { 3 }$ , $y = \frac { 1 } { 3 }$ , $z = \frac { 1 } { 3 }$

B) $x = \frac { 11 } { 3 }$ , $y = \frac { 4 } { 3 }$ , $z = \frac { 1 } { 3 }$

C) $x = \frac { 11 } { 3 }$ , $y = \frac { 1 } { 3 }$ , $z = - \frac { 2 } { 3 }$

D) $x = 3$ , $y = 1$ , $z = 11$

E) no solution

Question 59

A boat travels upstream (against the current) between two points on a river $10$ mi apart in $2.5$ hours. The return trip downstream takes only $45$ minutes. What is the boat's speed, and how fast does the current in the river flow?

A)The speed of the boat is $4.5$ Mi/h and the speed of the current flow is about $8.4$ Mi/h
B)The speed of the boat is $8.7$ Mi/h and the speed of the current flow is about $4.7$ Mi/h
C)The speed of the boat is $6.7$ Mi/h and the speed of the current flow is about $8.7$ Mi/h
D)The speed of the boat is $5$ Mi/h and the speed of the current flow is about $4$ Mi/h
E)The speed of the boat is $13.7$ Mi/h and the speed of the current flow is about $6.7$ Mi/h.

Question 60

Given $A = \left[ \begin{array} { c c } - 9 & 8 \\- 6 & - 5 \\10 & - 16\end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \\2 & 3 \\0 & 4\end{array} \right]$

, solve $2 X - A = 3 B$ , for $X$

A) $X = \left[ \begin{array} { l l l } - 3 & 4 & 0 \\5 & 2 & 2\end{array} \right]$

B) $X = \left[ \begin{array} { l l } 5 & 4 \\0 & 1 \\2 & 2\end{array} \right]$

C) $X = \left[ \begin{array} { l l } 5 & \frac { 1 } { 2 } \\0 & 2 \\5 & 2\end{array} \right]$

D) $X = \left[ \begin{array} { c c } - 3 & 4 \\0 & 2 \\5 & - 2\end{array} \right]$

E) none

Question 61

Solve the system of equations by converting to a matrix equation and using its inverse coefficient matrix

A)
B)
C)
D)
E)

Question 62

A triangle has vertices $\left( \frac { 3 } { 2 } , 0 \right)$ , $( 2,6 )$ , and $( 1,4 )$ .
Use a determinant to find its area

A) $\frac { 4 } { 3 }$

B) $\frac { 5 } { 3 }$

C) $\frac { 5 } { 2 }$

D) $\frac { 2 } { 5 }$

E) $2$

Question 63

Solve the system.

A)
B)
C)
D)
E)

Question 64

Use Cramer's Rule to solve the system.

A)
B)
C)
D)
E)

Question 65

Find the dimension of matrix A.

A)
B)
C)
D)
E)none

Question 66

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

A) $\left[ \begin{array} { c c c } 2 & - 2 & 0 \\0 & 2 & 0 \\- 2 & 1 / 2 & 1\end{array} \right]$

B) $\left[ \begin{array} { c c c } 2 & - 1 / 2 & 0 \\1 & 2 & 0 \\- 2 & 1 / 2 & 1\end{array} \right]$

C) $\left[ \begin{array} { c c c } 1 & - 1 & 1 \\0 & 1 / 2 & 0 \\- 1 & 1 / 2 & 1\end{array} \right]$

D) $\left[ \begin{array} { c c c } 1 / 2 & - 1 / 2 & 0 \\0 & 1 / 2 & 0 \\- 1 / 2 & 1 / 2 & 1\end{array} \right]$

E) none

Question 67

Solve $\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right]$ for $x$
And $y$

A) $x = - 2$ , $y = - 1$

B) $x = - \frac { 3 } { 2 }$ , $y = - 2$

C) $x = - \frac { 3 } { 2 }$ , $y = - 1$

D) $x = \frac { J } { 2 }$ , $y = - \frac { 3 } { 2 }$

E) none

Question 68

Find the partial fraction decomposition of

A)
B)
C)
E)none
D)

Question 69

Which point is a solution to the inequality ?

A)
B)
C)
D)
E)

Question 70

Find the determinant of the matrix $\left[ \begin{array} { r r } \frac { 1 } { 2 } & 1 \\\\- \frac { 1 } { 2 } & \frac { 3 } { 2 }\end{array} \right]$ , if it exists

A) $\frac { 3 } { 4 }$
B) $5$
C) $- \frac { 5 } { 4 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 5 } { 4 }$

Question 71

The system is linear. $\left\{ \begin{array} { r } x + 2 y + z = 3 \\\sqrt { 2 } x + y - \sqrt { 3 } z = 1 \\x + y + 4 z = - 4\end{array} \right.$

Question 72

Determine which point is a solution for the system of inequalities.

A)
B)
C)
D)
E)

Question 73

Write the system of equations in matrix form.

A)
B)
C)
E) none
D)

Question 74

Given $A = \left[ \begin{array} { l l } 0 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 1 & 5 \\- 2 & 3\end{array} \right]$ , $C = \left[ \begin{array} { l l l } 0 & 1 & 2 \\6 & 2 & 3\end{array} \right]$ , $D = \left[ \begin{array} { l } 1 \\5 \\0\end{array} \right]$

, find $( A B ) \cdot ( C D )$

A) $190$
B) $150$
C) $45$
D) $- 150$
E) The operation cannot be performed

Question 75

Given $A = \left[ \begin{array} { c c c } 2 & 3 & - 1 \\4 & - 2 & 0 \\- 3 & - 5 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 3 & 5 & - 3 \\5 & 2 & 2 \\- 7 & 4 & 2\end{array} \right]$
, find $3 A - 2 B$

A) $\left[ \begin{array} { c c c } 2 & 1 & 3 \\2 & 10 & - 4 \\5 & 3 & 2\end{array} \right]$

B) $\left[ \begin{array} { c c c } 1 & 1 & 3 \\3 & 10 & - 4 \\4 & 23 & 5\end{array} \right]$

C) $\left[ \begin{array} { c c c } 0 & - 1 & 3 \\2 & - 10 & 4 \\5 & - 23 & 2\end{array} \right]$

D) $\left[ \begin{array} { c c c } 2 & - 1 & - 3 \\0 & 10 & 4 \\5 & - 14 & - 2\end{array} \right]$

E) none

Question 76

Write out the form of the partial fraction decomposition of $\frac { x ^ { 2 } + 5 } { x ^ { 3 } - 2 x ^ { 2 } + 4 x - 8 }$

A) $\frac { A } { x - 2 } + \frac { B x } { ( x - 2 ) ^ { 2 } } + \frac { C } { x + 2 }$

B) $\frac { A x + B } { x - 2 } + \frac { C } { x ^ { 2 } + 4 }$

C) $\frac { A } { x - 2 } + \frac { B x } { ( x - 2 ) ^ { 2 } }$

D) $\frac { A } { x - 2 } + \frac { B x + C } { x ^ { 2 } + 4 }$

E)none

Question 77

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 3,7 )$ , $( - 2,1 )$ , and $( 1,7 )$

A) $y = x ^ { 2 } + x$
B) $y = x ^ { 2 } - 4 x + 2$
C) $y = x ^ { 2 } + 2 x - 1$
D) $y = 2 x ^ { 2 } + 4$
E) $y = 2 x ^ { 2 } + 4 x + 1$

Question 78

Given $A = \left[ \begin{array} { l l } 0 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 1 & 5 \\- 2 & 3\end{array} \right]$ , $C = \left[ \begin{array} { l } 1 \\5 \\0\end{array} \right]$ , $D = \left[ \begin{array} { l l l } 0 & 1 & 2 \\6 & 2 & 3\end{array} \right]$ , find $( A B ) \cdot ( C D )$

A) $190$
B) $150$
C) $45$
D) $- 150$
E) The operation cannot be performed

Question 79

Write out the form of the partial fraction decomposition of

A)
B)
C)
D)
E)none

Question 80

Use Cramer's Rule to solve the system.

A)
B)
C)
D)
E)no solution