Question 1

Solve the equation Ax = b by using the LU factorization given for A.
-$$\begin{array} { l } 
A = \left[ \begin{array} { r r r } 
3 & - 1 & 2 \
- 6 & 4 & - 5 \
9 & 5 & 6
\end{array} \right] , \mathbf { b } = \left[ \begin{array} { r } 
6 \
- 3 \
2
\end{array} \right] \
A = \left[ \begin{array} { r r r } 
1 & 0 & 0 \
- 2 & 1 & 0 \
3 & 4 & 1
\end{array} \right] \left[ \begin{array} { r r r } 
3 & - 1 & 2 \
0 & 2 & - 1 \
0 & 0 & 4
\end{array} \right]
\end{array}$$

Accepted Answer

A)  $\mathbf { x } = \left[ \begin{array} { r } 49 \ - 38 \ 32 \end{array} \right]$ 
B)  $\mathbf { x } = \left[ \begin{array} { c } 22 \ - 7 \ 15 \end{array} \right]$ 
C)  $\mathbf { x } = \left[ \begin{array} { c } 10 \ - 2 \ - 13 \end{array} \right]$ 
D)  $\mathbf { x } = \left[ \begin{array} { r } 25 \ - 58 \ 51 \end{array} \right]$ 
A)  $\mathbf { x } = \left[ \begin{array} { r } 49 \ - 38 \ 32 \end{array} \right]$ 
B)  $\mathbf { x } = \left[ \begin{array} { c } 22 \ - 7 \ 15 \end{array} \right]$ 
C)  $\mathbf { x } = \left[ \begin{array} { c } 10 \ - 2 \ - 13 \end{array} \right]$ 
D)  $\mathbf { x } = \left[ \begin{array} { r } 25 \ - 58 \ 51 \end{array} \right]$ C

Question 2

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
2 x _ { 1 } - 4 x _ { 2 } = - 2 \
3 x _ { 1 } + 4 x _ { 2 } = - 23
\end{array}$$

Accepted Answer

A)  $( - 2,5 )$ 
B)  $( 5,2 )$ 
C)  $( 2,5 )$ 
D)  $( - 5 , - 2 )$ 
A)  $( - 2,5 )$ 
B)  $( 5,2 )$ 
C)  $( 2,5 )$ 
D)  $( - 5 , - 2 )$ D

Question 3

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are
defined.
-$A$ is $2 \times 1$, B is $1 \times 1$.

Accepted Answer

A)  $\mathrm { AB }$ is $2 \times 1 , \mathrm { BA }$ is undefined. 
B)  $\mathrm { AB }$ is $1 \times 2 , \mathrm { BA }$ is $1 \times 1$. 
C)  $\mathrm { AB }$ is $2 \times 2 , \mathrm { BA }$ is $1 \times 1$. 
D)  $A B$ is undefined, $B A$ is $1 \times 2$. 
A)  $\mathrm { AB }$ is $2 \times 1 , \mathrm { BA }$ is undefined. 
B)  $\mathrm { AB }$ is $1 \times 2 , \mathrm { BA }$ is $1 \times 1$. 
C)  $\mathrm { AB }$ is $2 \times 2 , \mathrm { BA }$ is $1 \times 1$. 
D)  $A B$ is undefined, $B A$ is $1 \times 2$. A

Question 4

Find the 4 × 4 matrix that produces the described transformation, using homogeneous coordinates.
-Rotation about the y-axis through an angle of 60° 73)

Accepted Answer

A) 
$$\left[ \begin{array} { c c c c } 
0.5 & 0 & \sqrt { 3 } / 2 & 0 \
0 & 1 & 0 & 0 \
- \sqrt { 3 } / 2 & 0 & 0.5 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { c c c c } 
\sqrt { 3 } / 2 & 0 & 0.5 & 0 \
0 & 1 & 0 & 0 \
- 0.5 & 0 & \sqrt { 3 } / 2 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c c } 
0.5 & \sqrt { 3 } / 2 & 0 & 0 \
- \sqrt { 3 } / 2 & 0.5 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c c c } 
1 & 0 & 0 & 0 \
0 & 0.5 & \sqrt { 3 } / 2 & 0 \
0 & - \sqrt { 3 } / 2 & 0.5 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c c c } 
0.5 & 0 & \sqrt { 3 } / 2 & 0 \
0 & 1 & 0 & 0 \
- \sqrt { 3 } / 2 & 0 & 0.5 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { c c c c } 
\sqrt { 3 } / 2 & 0 & 0.5 & 0 \
0 & 1 & 0 & 0 \
- 0.5 & 0 & \sqrt { 3 } / 2 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c c } 
0.5 & \sqrt { 3 } / 2 & 0 & 0 \
- \sqrt { 3 } / 2 & 0.5 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c c c } 
1 & 0 & 0 & 0 \
0 & 0.5 & \sqrt { 3 } / 2 & 0 \
0 & - \sqrt { 3 } / 2 & 0.5 & 0 \
0 & 0 & 0 & 1
\end{array} \right]$$

Question 5

Decide whether or not the matrices are inverses of each other.
-$\left[ \begin{array} { r r } - 5 & - 1 \ 6 & 0 \end{array} \right]$ and $\left[ \begin{array} { c } 0 \frac { 1 } { 6 } \ - 1 \frac { 5 } { 6 } \end{array} \right]$

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 6

Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates.
-Translate by (8, 6), and then reflect through the line y = x.

Accepted Answer

A) 
$\left[\begin{array}{rrr}
-1 & 0 & -8 \
0 & -1 & -6 \
0 & 0 & 1
\end{array}\right]$ 
B) 
$\left[\begin{array}{lll}
0 & 1 & 8 \
1 & 0 & 6 \
0 & 0 & 1
\end{array}\right]$ 
C) 
$\left[\begin{array}{lll}
0 & 6 & 1 \
8 & 0 & 0 \
0 & 0 & 1
\end{array}\right]$ 
D) 
$\left[ \begin{array} { l l l } 0 & 1 & 6 \ 1 & 0 & 8 \ 0 & 0 & 1 \end{array} \right]$ 
A) 
$\left[\begin{array}{rrr}
-1 & 0 & -8 \
0 & -1 & -6 \
0 & 0 & 1
\end{array}\right]$ 
B) 
$\left[\begin{array}{lll}
0 & 1 & 8 \
1 & 0 & 6 \
0 & 0 & 1
\end{array}\right]$ 
C) 
$\left[\begin{array}{lll}
0 & 6 & 1 \
8 & 0 & 0 \
0 & 0 & 1
\end{array}\right]$ 
D) 
$\left[ \begin{array} { l l l } 0 & 1 & 6 \ 1 & 0 & 8 \ 0 & 0 & 1 \end{array} \right]$

Question 7

Find the transpose of the matrix.
-$\left[ \begin{array} { r r r r } 7 & 4 & 7 & 4 \ 0 & - 7 & 0 & - 7 \end{array} \right]$

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
0 & 7 \
- 7 & 4 \
0 & 7 \
- 7 & 4
\end{array} \right]$$ 
B) 
$\left[\begin{array}{rr}
7 & 0 \
4 & -7 \
7 & 0 \
4 & -7
\end{array}\right]$ 
C) 
$\left[\begin{array}{rrrr}
0 & -7 & 0 & -7 \
7 & 4 & 7 & 4
\end{array}\right]$ 
D) 
$\left[\begin{array}{rrrr}
4 & 7 & 4 & 7 \
-7 & 0 & -7 & 0
\end{array}\right]$ 
A) 
$$\left[ \begin{array} { r r } 
0 & 7 \
- 7 & 4 \
0 & 7 \
- 7 & 4
\end{array} \right]$$ 
B) 
$\left[\begin{array}{rr}
7 & 0 \
4 & -7 \
7 & 0 \
4 & -7
\end{array}\right]$ 
C) 
$\left[\begin{array}{rrrr}
0 & -7 & 0 & -7 \
7 & 4 & 7 & 4
\end{array}\right]$ 
D) 
$\left[\begin{array}{rrrr}
4 & 7 & 4 & 7 \
-7 & 0 & -7 & 0
\end{array}\right]$

Question 8

Solve the system by using the inverse of the coefficient matrix.
-10x1 - 4x2 = -6 6x1 - x2 = 2

Accepted Answer

A) (4, 1) 
B) (1, 4) 
C) (-1, -4) 
D) (-4, -1) 
A) (4, 1) 
B) (1, 4) 
C) (-1, -4) 
D) (-4, -1)

Question 9

Determine whether b is in the column space of A.
-$$A = \left[ \begin{array} { r r c } 
- 1 & 0 & 2 \
5 & 8 & - 10 \
- 3 & - 3 & 6
\end{array} \right] , \mathbf { b } = \left[ \begin{array} { r } 
- 4 \
3 \
4
\end{array} \right]$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 10

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
2 x _ { 1 } + 6 x _ { 2 } = 2 \
2 x _ { 1 } - x _ { 2 } = - 5
\end{array}$$

Accepted Answer

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

Question 11

Find the matrix product AB, if it is defined.
-$$A = \left[ \begin{array} { r r } 
0 & - 3 \
4 & 3
\end{array} \right] , B = \left[ \begin{array} { l l } 
- 2 & 0 \
- 1 & 1
\end{array} \right]$$

Accepted Answer

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

Question 12

Perform the matrix operation.
-Let $B = \left[ \begin{array} { l l l l } - 1 & 1 & 7 & - 3 \end{array} \right]$. Find $- 4 B$.

Accepted Answer

A)  $\left[ \begin{array} { l l l l } - 4 & 4 & 28 & - 12 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l l } 4 & 1 & 7 & - 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l l } - 3 & - 1 & 5 & - 5 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } 4 & - 4 & - 28 & 12 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l l l } - 4 & 4 & 28 & - 12 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l l } 4 & 1 & 7 & - 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l l } - 3 & - 1 & 5 & - 5 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } 4 & - 4 & - 28 & 12 \end{array} \right]$

Question 13

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
2 x _ { 1 } - 6 x _ { 2 } = - 6 \
3 x _ { 1 } + 2 x _ { 2 } = 13
\end{array}$$

Accepted Answer

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

Question 14

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { l l } 
9 & 4 \
4 & 4
\end{array} \right] \text { and } \left[ \begin{array} { r r } 
- 0.2 & 0.2 \
0.2 & - 0.45
\end{array} \right]$$

Accepted Answer

A)  No 
B)  Yes 
A)  No 
B)  Yes

Question 15

Find the inverse of the matrix, if it exists
-$$A = \left[ \begin{array} { r r } 
1 & 4 \
0 & - 6
\end{array} \right]$$

Accepted Answer

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

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

$$\left[ \begin{array} { r r } 
- \frac { 1 } { 6 } & \frac { 2 } { 3 } \
0 & 1
\end{array} \right]$$

Question 16

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{aligned}
6 x _ { 1 } + 3 x _ { 2 } & = 0 \
2 x _ { 1 } & = - 6
\end{aligned}$$

Accepted Answer

A)  $( 6 , - 3 )$ 
B)  $( - 3,6 )$ 
C)  No solution 
D)  $( - 3 , - 6 )$ 
A)  $( 6 , - 3 )$ 
B)  $( - 3,6 )$ 
C)  No solution 
D)  $( - 3 , - 6 )$

Question 17

Find the inverse of the matrix, if it exists
-$$A = \left[ \begin{array} { l l } 
6 & 3 \
3 & 0
\end{array} \right]$$

Accepted Answer

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

Question 18

Decide whether or not the matrices are inverses of each other.
-$\left[ \begin{array} { r r } 6 & - 5 \ - 3 & 5 \end{array} \right]$ and $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 1 } { 3 } \ \frac { 1 } { 5 } & \frac { 2 } { 5 } \end{array} \right]$

Accepted Answer

A)  $\mathrm { No }$ 
B)  Yes 
A)  $\mathrm { No }$ 
B)  Yes

Question 19

Find the matrix product AB, if it is defined.
-$$A = \left[ \begin{array} { r r } 
3 & - 2 \
3 & 0
\end{array} \right] , B = \left[ \begin{array} { r r } 
0 & - 2 \
4 & 6
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { r r } - 18 & - 8 \ - 6 & 0 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } 0 & 4 \ 12 & 0 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } - 8 & - 18 \ 0 & - 6 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 6 & 0 \ 30 & - 8 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r } - 18 & - 8 \ - 6 & 0 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } 0 & 4 \ 12 & 0 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } - 8 & - 18 \ 0 & - 6 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 6 & 0 \ 30 & - 8 \end{array} \right]$

Question 20

Determine the rank of the matrix.
-$$\left[ \begin{array} { r r r r } 
1 & - 2 & 2 & - 3 \
2 & - 4 & 7 & - 2 \
- 3 & 6 & - 6 & 9
\end{array} \right]$$

Accepted Answer

A) 1 
B) 3 
C) 4 
D) 2 
A) 1 
B) 3 
C) 4 
D) 2

Solve the equation Ax = b by using the LU factorization given for A. - A= 3 -1 2 -6 4 -5 9 5 6 ,= 6 -3 2 A= 1 0 0 -2 1 0 3 4 1 3 -1 2 0 2 -1 0 0 4

Solve the system by using the inverse of the coefficient matrix. - 2-4=-2 3+4=-23

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. - $A$ is $2 \times 1$ , B is $1 \times 1$ .

Find the 4 × 4 matrix that produces the described transformation, using homogeneous coordinates. -Rotation about the y-axis through an angle of 60° 73)

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } - 5 & - 1 \\ 6 & 0 \end{array} \right]$ and $\left[ \begin{array} { c } 0 \frac { 1 } { 6 } \\ - 1 \frac { 5 } { 6 } \end{array} \right]$

Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates. -Translate by (8, 6), and then reflect through the line y = x.

Find the transpose of the matrix. - $\left[ \begin{array} { r r r r } 7 & 4 & 7 & 4 \\ 0 & - 7 & 0 & - 7 \end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. -10x1 - 4x2 = -6 6x1 - x2 = 2

Determine whether b is in the column space of A. - $A = \left[ \begin{array} { r r c } - 1 & 0 & 2 \\5 & 8 & - 10 \\- 3 & - 3 & 6\end{array} \right] , \mathbf { b } = \left[ \begin{array} { r } - 4 \\3 \\4\end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. - 2+6=2 2-=-5

Find the matrix product AB, if it is defined. - $A = \left[ \begin{array} { r r } 0 & - 3 \\4 & 3\end{array} \right] , B = \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right]$

Perform the matrix operation. -Let $B = \left[ \begin{array} { l l l l } - 1 & 1 & 7 & - 3 \end{array} \right]$ . Find $- 4 B$ .

Solve the system by using the inverse of the coefficient matrix. - 2-6=-6 3+2=13

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 9 & 4 \\4 & 4\end{array} \right] \text { and } \left[ \begin{array} { r r } - 0.2 & 0.2 \\0.2 & - 0.45\end{array} \right]$

Find the inverse of the matrix, if it exists - $A = \left[ \begin{array} { r r } 1 & 4 \\0 & - 6\end{array} \right]$

Solve the system by using the inverse of the coefficient matrix. - 6+3 =0 2 =-6

Find the inverse of the matrix, if it exists - $A = \left[ \begin{array} { l l } 6 & 3 \\3 & 0\end{array} \right]$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } 6 & - 5 \\ - 3 & 5 \end{array} \right]$ and $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 1 } { 3 } \\ \frac { 1 } { 5 } & \frac { 2 } { 5 } \end{array} \right]$

Find the matrix product AB, if it is defined. - $A = \left[ \begin{array} { r r } 3 & - 2 \\3 & 0\end{array} \right] , B = \left[ \begin{array} { r r } 0 & - 2 \\4 & 6\end{array} \right]$

Determine the rank of the matrix. - $\left[ \begin{array} { r r r r } 1 & - 2 & 2 & - 3 \\2 & - 4 & 7 & - 2 \\- 3 & 6 & - 6 & 9\end{array} \right]$

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Exam 2: Matrix Algebra