Question 1

Find a basis for the column space of the matrix.
-$$B = \left[ \begin{array} { r r r r } 
1 & - 2 & 5 & - 3 \
2 & - 4 & 13 & - 2 \
- 3 & 6 & - 15 & 9
\end{array} \right]$$

Accepted Answer

A) 
$$
\left\{\left[\begin{array}{r}
1 \
2 \
-3
\end{array}\right],\left[\begin{array}{r}
5 \
13 \
-15
\end{array}\right]\right\}
$$ 
B)  $$
\left\{\left[\begin{array}{l}
1 \
0 \
0
\end{array}\right],\left[\begin{array}{l}
0 \
1 \
0
\end{array}\right]\right\}
$$ 
C)

$$
\left\{\left[\begin{array}{r}
1 \
2 \
-3
\end{array}\right],\left[\begin{array}{r}
-2 \
-4 \
6
\end{array}\right]\right\}
$$ 
D) 
$$
\left\{\left[\begin{array}{l}
2 \
1 \
0 \
0
\end{array}\right],\left[\begin{array}{c}
\frac{29}{3} \
0 \
-\frac{4}{3} \
1
\end{array}\right]\right\}
$$ 
A) 
$$
\left\{\left[\begin{array}{r}
1 \
2 \
-3
\end{array}\right],\left[\begin{array}{r}
5 \
13 \
-15
\end{array}\right]\right\}
$$ 
B)  $$
\left\{\left[\begin{array}{l}
1 \
0 \
0
\end{array}\right],\left[\begin{array}{l}
0 \
1 \
0
\end{array}\right]\right\}
$$ 
C)

$$
\left\{\left[\begin{array}{r}
1 \
2 \
-3
\end{array}\right],\left[\begin{array}{r}
-2 \
-4 \
6
\end{array}\right]\right\}
$$ 
D) 
$$
\left\{\left[\begin{array}{l}
2 \
1 \
0 \
0
\end{array}\right],\left[\begin{array}{c}
\frac{29}{3} \
0 \
-\frac{4}{3} \
1
\end{array}\right]\right\}
$$

Question 2

Find the inverse of the matrix, if it exists
-$$A = \left[ \begin{array} { r r } 
5 & 0 \
- 4 & - 6
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { c c } 
- \frac { 1 } { 6 } & 0 \
- \frac { 2 } { 15 } & \frac { 1 } { 5 }
\end{array} \right]$$ 
B)  $\mathrm { A }$ is not invertible 
C) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 5 } & 0 \
\frac { 2 } { 15 } & - \frac { 1 } { 6 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 5 } & 0 \
- \frac { 2 } { 15 } & - \frac { 1 } { 6 }
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { c c } 
- \frac { 1 } { 6 } & 0 \
- \frac { 2 } { 15 } & \frac { 1 } { 5 }
\end{array} \right]$$ 
B)  $\mathrm { A }$ is not invertible 
C) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 5 } & 0 \
\frac { 2 } { 15 } & - \frac { 1 } { 6 }
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c } 
\frac { 1 } { 5 } & 0 \
- \frac { 2 } { 15 } & - \frac { 1 } { 6 }
\end{array} \right]$$

Question 3

Find the inverse of the matrix A, if it exists.
-$$A = \left[ \begin{array} { r r r } 
5 & - 1 & 5 \
5 & 0 & 3 \
10 & - 1 & 8
\end{array} \right]$$

Accepted Answer

A)  $A ^ { - 1 } = \left[ \begin{array} { c c c } 1 & 0 & \frac { 3 } { 5 } \ 0 & 1 & - 2 \ 0 & \frac { 4 } { 5 } & 0 \end{array} \right]$ 
B)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 1 & 0 & \frac { 3 } { 5 } \ 0 & 1 & - 2 \ 0 & 0 & 0 \end{array} \right]$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 5 & 5 & 10 \ - 1 & 0 & - 1 \ 5 & 3 & 8 \end{array} \right]$ 
D)  $A ^ { - 1 }$ does not exist. 
A)  $A ^ { - 1 } = \left[ \begin{array} { c c c } 1 & 0 & \frac { 3 } { 5 } \ 0 & 1 & - 2 \ 0 & \frac { 4 } { 5 } & 0 \end{array} \right]$ 
B)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 1 & 0 & \frac { 3 } { 5 } \ 0 & 1 & - 2 \ 0 & 0 & 0 \end{array} \right]$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 5 & 5 & 10 \ - 1 & 0 & - 1 \ 5 & 3 & 8 \end{array} \right]$ 
D)  $A ^ { - 1 }$ does not exist.

Question 4

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { l l } 
9 & - 2 \
7 & - 2
\end{array} \right] \text { and } \left[ \begin{array} { c c } 
0.5 & 0.5 \
- \frac { 7 } { 4 } & - \frac { 9 } { 4 }
\end{array} \right]$$

Accepted Answer

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

Question 5

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

Accepted Answer

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

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

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

Question 6

Perform the matrix operation.
-Let $A = \left[ \begin{array} { c c } - 2 & 3 \ - 7 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { c c } 2 & 10 \ - 7 & 6 \end{array} \right]$. Find $A - B$.

Accepted Answer

A) 
$\left[ \begin{array} { c c } 4 & - 7 \ - 14 & 4 \end{array} \right]$ 
B) 
$\left[ \begin{array} { c c } - 4 & - 7 \ 0 & - 8 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 0 & - 7 \ 0 & - 8 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c c } 0 & 7 \ - 14 & 8 \end{array} \right]$ 
A) 
$\left[ \begin{array} { c c } 4 & - 7 \ - 14 & 4 \end{array} \right]$ 
B) 
$\left[ \begin{array} { c c } - 4 & - 7 \ 0 & - 8 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 0 & - 7 \ 0 & - 8 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c c } 0 & 7 \ - 14 & 8 \end{array} \right]$

Question 7

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

Accepted Answer

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

Question 8

Find the inverse of the matrix, if it exists
-$$A = \left[ \begin{array} { r r } 
- 3 & - 4 \
3 & - 4
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
- \frac { 1 } { 8 } & \frac { 1 } { 6 } \
- \frac { 1 } { 8 } & - \frac { 1 } { 6 }
\end{array} \right]$$ 
B) 
$\left[\begin{array}{cc}
-\frac{1}{6} & \frac{1}{6} \
-\frac{1}{8} & -\frac{1}{8}
\end{array}\right]$ 
C) 
  
D) 
$\left[\begin{array}{c}
-\frac{1}{6}-\frac{1}{6} \
\frac{1}{8}-\frac{1}{8}
\end{array}\right]$ 
A) 
$$\left[ \begin{array} { r r } 
- \frac { 1 } { 8 } & \frac { 1 } { 6 } \
- \frac { 1 } { 8 } & - \frac { 1 } { 6 }
\end{array} \right]$$ 
B) 
$\left[\begin{array}{cc}
-\frac{1}{6} & \frac{1}{6} \
-\frac{1}{8} & -\frac{1}{8}
\end{array}\right]$ 
C) 
  
D) 
$\left[\begin{array}{c}
-\frac{1}{6}-\frac{1}{6} \
\frac{1}{8}-\frac{1}{8}
\end{array}\right]$

Question 9

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

Accepted Answer

A) 
$\left[ \begin{array} { r r } 2 & - 6 \ - 1 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 6 & - 1 \ 4 & - 6 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } - 1 & 6 \ - 6 & 4 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r } 2 & 0 \ - 2 & 4 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } 2 & - 6 \ - 1 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 6 & - 1 \ 4 & - 6 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } - 1 & 6 \ - 6 & 4 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r } 2 & 0 \ - 2 & 4 \end{array} \right]$

Question 10

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
6 x _ { 1 } + 5 x _ { 2 } = 13 \
5 x _ { 1 } + 3 x _ { 2 } = 5
\end{array}$$

Accepted Answer

A)  $( 5 , - 2 )$ 
B)  $( - 2,5 )$ 
C)  No solution 
D)  $( - 2 , - 5 )$ 
A)  $( 5 , - 2 )$ 
B)  $( - 2,5 )$ 
C)  No solution 
D)  $( - 2 , - 5 )$

Question 11

Perform the matrix operation.
-Let $\mathrm { A } = \left[ \begin{array} { l l } - 1 & 2 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$. Find $3 \mathrm {~A} + 4 \mathrm {~B}$.

Accepted Answer

A)  $\left[ \begin{array} { l l } - 1 & 4 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } - 3 & 4 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 2 & 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 1 & 6 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l } - 1 & 4 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l l } - 3 & 4 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 2 & 2 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 1 & 6 \end{array} \right]$

Question 12

Find an LU factorization of the matrix A.
-$$A = \left[ \begin{array} { r r r } 
2 & 3 & 5 \
4 & 9 & 5 \
4 & - 3 & 24
\end{array} \right]$$

Accepted Answer

A)  $\mathrm { A } = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 4 & 1 & 0 \ 4 & - 3 & 1 \end{array} \right] \left[ \begin{array} { r r r } 2 & 3 & 5 \ 0 & 3 & - 5 \ 0 & 0 & - 1 \end{array} \right]$ 
B)  $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 2 & 1 & 0 \ 2 & - 3 & 1 \end{array} \right] \left[ \begin{array} { r r r } 3 & 3 & 5 \ 0 & - 3 & 5 \ 0 & 0 & 1 \end{array} \right]$ 
C)  $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 2 & 1 & 0 \ 2 & - 3 & 1 \end{array} \right] \left[ \begin{array} { c c r } 2 & 3 & 5 \ 0 & 3 & - 5 \ 0 & 0 & - 1 \end{array} \right]$ 
D)  $\mathrm { A } = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 4 & 1 & 0 \ 4 & - 3 & 1 \end{array} \right] \left[ \begin{array} { r r r } 2 & 3 & 5 \ 0 & 9 & 5 \ 0 & 0 & 24 \end{array} \right]$ 
A)  $\mathrm { A } = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 4 & 1 & 0 \ 4 & - 3 & 1 \end{array} \right] \left[ \begin{array} { r r r } 2 & 3 & 5 \ 0 & 3 & - 5 \ 0 & 0 & - 1 \end{array} \right]$ 
B)  $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 2 & 1 & 0 \ 2 & - 3 & 1 \end{array} \right] \left[ \begin{array} { r r r } 3 & 3 & 5 \ 0 & - 3 & 5 \ 0 & 0 & 1 \end{array} \right]$ 
C)  $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 2 & 1 & 0 \ 2 & - 3 & 1 \end{array} \right] \left[ \begin{array} { c c r } 2 & 3 & 5 \ 0 & 3 & - 5 \ 0 & 0 & - 1 \end{array} \right]$ 
D)  $\mathrm { A } = \left[ \begin{array} { r r r } 1 & 0 & 0 \ 4 & 1 & 0 \ 4 & - 3 & 1 \end{array} \right] \left[ \begin{array} { r r r } 2 & 3 & 5 \ 0 & 9 & 5 \ 0 & 0 & 24 \end{array} \right]$

Question 13

Perform the matrix operation.
-Let $A = \left[ \begin{array} { c c } - 3 & 2 \ 3 & - 5 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 0 & 0 \ 0 & 0 \end{array} \right]$. Find $A + B$.

Accepted Answer

A)  Undefined 
B) 
$\left[ \begin{array} { l l } 0 & 0 \ 0 & 0 \end{array} \right]$ 
C) 
$$\left[ \begin{array} { c c } 
-3 & 2 \
 3 & -5
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c } 
3 & - 2 \
- 3 & 5
\end{array} \right]$$ 
A)  Undefined 
B) 
$\left[ \begin{array} { l l } 0 & 0 \ 0 & 0 \end{array} \right]$ 
C) 
$$\left[ \begin{array} { c c } 
-3 & 2 \
 3 & -5
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { c c } 
3 & - 2 \
- 3 & 5
\end{array} \right]$$

Question 14

Find the matrix product AB, if it is defined.
-$$A = \left[ \begin{array} { r r r } 
1 & 3 & - 1 \
3 & 0 & 5
\end{array} \right] , B = \left[ \begin{array} { r r } 
3 & 0 \
- 1 & 1 \
0 & 5
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { l l } 0 & - 2 \ 9 & 25 \end{array} \right]$ 
B)

$\left[ \begin{array} { c c } - 2 & 0 \ 25 & 9 \end{array} \right]$ 
C)  $\mathrm { AB }$ is undefined. 
D) 
$$\left[ \begin{array} { r r r } 
3 & - 3 & 0 \
0 & 0 & 25
\end{array} \right]$$ 
A) 
$\left[ \begin{array} { l l } 0 & - 2 \ 9 & 25 \end{array} \right]$ 
B)

$\left[ \begin{array} { c c } - 2 & 0 \ 25 & 9 \end{array} \right]$ 
C)  $\mathrm { AB }$ is undefined. 
D) 
$$\left[ \begin{array} { r r r } 
3 & - 3 & 0 \
0 & 0 & 25
\end{array} \right]$$

Question 15

Find the inverse of the matrix A, if it exists.
-$$A = \left[ \begin{array} { l l l } 
1 & 0 & 8 \
1 & 2 & 3 \
2 & 5 & 3
\end{array} \right]$$

Accepted Answer

A)  $A ^ { - 1 } = \left[ \begin{array} { r r r } - 1 & 0 & - 8 \ - 1 & - 2 & - 3 \ - 2 & - 5 & - 3 \end{array} \right]$ 
B)  $\mathrm { A } ^ { - 1 }$ does not exist. 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 9 & - 40 & 16 \ - 3 & 13 & - 5 \ - 1 & 5 & - 2 \end{array} \right]$ 
D)  $\mathrm { A } ^ { - 1 } = \left[ \begin{array} { l l l } 1 & 1 & 2 \ 0 & 2 & 5 \ 8 & 3 & 3 \end{array} \right]$ 
A)  $A ^ { - 1 } = \left[ \begin{array} { r r r } - 1 & 0 & - 8 \ - 1 & - 2 & - 3 \ - 2 & - 5 & - 3 \end{array} \right]$ 
B)  $\mathrm { A } ^ { - 1 }$ does not exist. 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } 9 & - 40 & 16 \ - 3 & 13 & - 5 \ - 1 & 5 & - 2 \end{array} \right]$ 
D)  $\mathrm { A } ^ { - 1 } = \left[ \begin{array} { l l l } 1 & 1 & 2 \ 0 & 2 & 5 \ 8 & 3 & 3 \end{array} \right]$

Question 16

Find the inverse of the matrix A, if it exists.
-$$A = \left[ \begin{array} { r r r } 
0 & 3 & 3 \
- 1 & 0 & 4 \
0 & 7 & 0
\end{array} \right]$$

Accepted Answer

A)  $A ^ { - 1 }$ does not exist. 
B)  $A ^ { - 1 } = \left[ \begin{array} { c c c } - \frac { 3 } { 3 } & - 1 & - \frac { 7 } { 7 } \ - \frac { 1 } { 7 } & 0 & \frac { 1 } { 7 } \ \frac { 1 } { 3 } & 0 & 0 \end{array} \right]$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } \frac { 4 } { 3 } & 0 & \frac { 1 } { 3 } \ - 1 & 0 & 0 \ - \frac { 4 } { 7 } & \frac { 1 } { 7 } & - \frac { 1 } { 7 } \end{array} \right]$ 
D)  $A ^ { - 1 } = \left[ \begin{array} { c c r } \frac { 4 } { 3 } & - 1 & - \frac { 4 } { 7 } \ 0 & 0 & \frac { 1 } { 7 } \ \frac { 1 } { 3 } & 0 & - \frac { 1 } { 7 } \end{array} \right]$ 
A)  $A ^ { - 1 }$ does not exist. 
B)  $A ^ { - 1 } = \left[ \begin{array} { c c c } - \frac { 3 } { 3 } & - 1 & - \frac { 7 } { 7 } \ - \frac { 1 } { 7 } & 0 & \frac { 1 } { 7 } \ \frac { 1 } { 3 } & 0 & 0 \end{array} \right]$ 
C)  $A ^ { - 1 } = \left[ \begin{array} { r r r } \frac { 4 } { 3 } & 0 & \frac { 1 } { 3 } \ - 1 & 0 & 0 \ - \frac { 4 } { 7 } & \frac { 1 } { 7 } & - \frac { 1 } { 7 } \end{array} \right]$ 
D)  $A ^ { - 1 } = \left[ \begin{array} { c c r } \frac { 4 } { 3 } & - 1 & - \frac { 4 } { 7 } \ 0 & 0 & \frac { 1 } { 7 } \ \frac { 1 } { 3 } & 0 & - \frac { 1 } { 7 } \end{array} \right]$

Question 17

Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and
final demand vector d. Round production levels to the nearest whole number.
-Compute the matrix of the transformation that performs the shear transformation $x \rightarrow A x$ for $A = \left[ \begin{array} { l l } 1 & 0.20 \ 0 & 1 \end{array} \right]$ and then scales all $x$-coordinates by a factor of $0.61$.

Accepted Answer

A) 
$\left[ \begin{array} { l l } 0.61 & 0.122 \ 0 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 1.61 & 0.20 \ 0 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 1 & 0.20 \ 0 & 0.61 \end{array} \right]$ 
D)

$\left[ \begin{array} { l l } 0.61 & 0.20 \ 0 & 1 \end{array} \right]$ 
A) 
$\left[ \begin{array} { l l } 0.61 & 0.122 \ 0 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 1.61 & 0.20 \ 0 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 1 & 0.20 \ 0 & 0.61 \end{array} \right]$ 
D)

$\left[ \begin{array} { l l } 0.61 & 0.20 \ 0 & 1 \end{array} \right]$

Question 18

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

Accepted Answer

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

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

$$\left[ \begin{array} { r r r } 
12 & - 8 & 4 \
- 6 & 12 & - 4
\end{array} \right]$$

Question 19

Determine whether the matrix is invertible.
-$$\left[ \begin{array} { r r } 
2 & 9 \
1 & 14
\end{array} \right]$$

Accepted Answer

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

Question 20

The vector x is in a subspace H with a basis β = {bS1U1B11S1U1B0, bS1U1B12S1U1B0}. Find the β-coordinate vector of x.

-$$\mathbf { b } _ { 1 } = \left[ \begin{array} { r } 
1 \
- 2
\end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { r } 
- 5 \
3
\end{array} \right] , \mathbf { x } = \left[ \begin{array} { r } 
22 \
- 16
\end{array} \right]$$

Accepted Answer

A)  $ \left[\begin{array}{r}-4 \ 1\end{array}\right] $ 
B)  $ \left[\begin{array}{r}-2 \ 4\end{array}\right] $ 
C) $ \left[\begin{array}{r}2 \ -4\end{array}\right] $ 
D) $ \left[\begin{array}{r}-4 \ 2\end{array}\right] $ 
A)  $ \left[\begin{array}{r}-4 \ 1\end{array}\right] $ 
B)  $ \left[\begin{array}{r}-2 \ 4\end{array}\right] $ 
C) $ \left[\begin{array}{r}2 \ -4\end{array}\right] $ 
D) $ \left[\begin{array}{r}-4 \ 2\end{array}\right] $

Find a basis for the column space of the matrix. - $B = \left[ \begin{array} { r r r r } 1 & - 2 & 5 & - 3 \\2 & - 4 & 13 & - 2 \\- 3 & 6 & - 15 & 9\end{array} \right]$

Find the inverse of the matrix, if it exists - $A = \left[ \begin{array} { r r } 5 & 0 \\- 4 & - 6\end{array} \right]$

Find the inverse of the matrix A, if it exists. - $A = \left[ \begin{array} { r r r } 5 & - 1 & 5 \\5 & 0 & 3 \\10 & - 1 & 8\end{array} \right]$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 9 & - 2 \\7 & - 2\end{array} \right] \text { and } \left[ \begin{array} { c c } 0.5 & 0.5 \\- \frac { 7 } { 4 } & - \frac { 9 } { 4 }\end{array} \right]$

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

Perform the matrix operation. -Let $A = \left[ \begin{array} { c c } - 2 & 3 \\ - 7 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { c c } 2 & 10 \\ - 7 & 6 \end{array} \right]$ . Find $A - B$ .

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

Find the inverse of the matrix, if it exists - $A = \left[ \begin{array} { r r } - 3 & - 4 \\3 & - 4\end{array} \right]$

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

Solve the system by using the inverse of the coefficient matrix. - 6+5=13 5+3=5

Perform the matrix operation. -Let $\mathrm { A } = \left[ \begin{array} { l l } - 1 & 2 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $3 \mathrm {~A} + 4 \mathrm {~B}$ .

Find an LU factorization of the matrix A. - $A = \left[ \begin{array} { r r r } 2 & 3 & 5 \\4 & 9 & 5 \\4 & - 3 & 24\end{array} \right]$

Perform the matrix operation. -Let $A = \left[ \begin{array} { c c } - 3 & 2 \\ 3 & - 5 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 0 & 0 \\ 0 & 0 \end{array} \right]$ . Find $A + B$ .

Find the matrix product AB, if it is defined. - $A = \left[ \begin{array} { r r r } 1 & 3 & - 1 \\3 & 0 & 5\end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\- 1 & 1 \\0 & 5\end{array} \right]$

Find the inverse of the matrix A, if it exists. - $A = \left[ \begin{array} { l l l } 1 & 0 & 8 \\1 & 2 & 3 \\2 & 5 & 3\end{array} \right]$

Find the inverse of the matrix A, if it exists. - $A = \left[ \begin{array} { r r r } 0 & 3 & 3 \\- 1 & 0 & 4 \\0 & 7 & 0\end{array} \right]$

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

Determine whether the matrix is invertible. - $\left[ \begin{array} { r r } 2 & 9 \\1 & 14\end{array} \right]$

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Exam 2: Matrix Algebra

Find a basis for the column space of the matrix. - B=[1−25−32−413−2−36−159]B = \left[ \begin{array} { r r r r } 1 & - 2 & 5 & - 3 \\2 & - 4 & 13 & - 2 \\- 3 & 6 & - 15 & 9\end{array} \right]B=​12−3​−2−46​513−15​−3−29​​

Find the inverse of the matrix, if it exists - A=[50−4−6]A = \left[ \begin{array} { r r } 5 & 0 \\- 4 & - 6\end{array} \right]A=[5−4​0−6​]

Find the inverse of the matrix A, if it exists. - A=[5−1550310−18]A = \left[ \begin{array} { r r r } 5 & - 1 & 5 \\5 & 0 & 3 \\10 & - 1 & 8\end{array} \right]A=​5510​−10−1​538​​

Find the inverse of the matrix, if it exists - [100−110111]\left[ \begin{array} { r r r } 1 & 0 & 0 \\- 1 & 1 & 0 \\1 & 1 & 1\end{array} \right]​1−11​011​001​​

Perform the matrix operation. -Let A=[−23−7−2]A = \left[ \begin{array} { c c } - 2 & 3 \\ - 7 & - 2 \end{array} \right]A=[−2−7​3−2​] and B=[210−76]B = \left[ \begin{array} { c c } 2 & 10 \\ - 7 & 6 \end{array} \right]B=[2−7​106​] . Find A−BA - BA−B .

Find the inverse of the matrix, if it exists - A=[−3−43−4]A = \left[ \begin{array} { r r } - 3 & - 4 \\3 & - 4\end{array} \right]A=[−33​−4−4​]

Find the matrix product AB, if it is defined. - A=[−1322],B=[−20−12]A = \left[ \begin{array} { r r } - 1 & 3 \\2 & 2\end{array} \right] , B = \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 2\end{array} \right]A=[−12​32​],B=[−2−1​02​]