Question 1

Use Cramer's rule to solve the system.
-$$\begin{aligned}
6 x - 7 y - z & = - 53 \
x - 8 y + 9 z & = 19 \
- 3 x + y + z & = 11
\end{aligned}$$

Accepted Answer

A)  $\{ ( 2,8,9 ) \}$ 
B)  $\{ ( 8,9,8 ) \}$ 
C)  $\{ ( 2 , - 8 , - 9 ) \}$ 
D)  $\{ ( 3,6,9 ) \}$ 
A)  $\{ ( 2,8,9 ) \}$ 
B)  $\{ ( 8,9,8 ) \}$ 
C)  $\{ ( 2 , - 8 , - 9 ) \}$ 
D)  $\{ ( 3,6,9 ) \}$

Question 2

Use Cramer's rule to solve the system.
-$$\begin{array} { l } 
2 x + 5 y = 43 \
2 x + 2 y = 22
\end{array}$$

Accepted Answer

A)  $\{ ( 4,7 ) \}$ 
B)  $\{ ( 7,4 ) \}$ 
C)  $\{ ( - 7,4 ) \}$ 
D)  $\{ ( 4 , - 7 ) \}$ 
A)  $\{ ( 4,7 ) \}$ 
B)  $\{ ( 7,4 ) \}$ 
C)  $\{ ( - 7,4 ) \}$ 
D)  $\{ ( 4 , - 7 ) \}$

Question 3

Solve the problem.
-Let $\mathrm { A } = \left[ \begin{array} { r r r } 7 & 2 & 4 \ 14 & 0 & - 1 \ 2 & 5 & 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 5 & - 2 & 1 \ 2 & 0 & 1 \ - 3 & 6 & - 5 \end{array} \right]$. Find $\mathrm { A } + \mathrm { B }$.

Accepted Answer

A)  $\left[ \begin{array} { r r r } 12 & 0 & 5 \ 16 & 0 & 0 \ - 1 & 11 & - 2 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 16 & - 4 & 5 \ 12 & 0 & 0 \ - 1 & 11 & - 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 16 & - 4 & 5 \ 16 & 0 & - 2 \ 1 & 11 & - 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } 12 & 0 & 5 \ 12 & 0 & - 2 \ 1 & 11 & - 2 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } 12 & 0 & 5 \ 16 & 0 & 0 \ - 1 & 11 & - 2 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 16 & - 4 & 5 \ 12 & 0 & 0 \ - 1 & 11 & - 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 16 & - 4 & 5 \ 16 & 0 & - 2 \ 1 & 11 & - 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } 12 & 0 & 5 \ 12 & 0 & - 2 \ 1 & 11 & - 2 \end{array} \right]$

Question 4

Solve the problem.
-Let $A = \left[ \begin{array} { r } 3 \ - 5 \ - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r } - 5 \ 6 \ 5 \end{array} \right]$. Find $A + B$.

Accepted Answer

A) 
$\left[\begin{array}{r}
-2 \
1 \
1
\end{array}\right]$ 
B) 
$\left[ \begin{array} { l l l } - 2 & 1 & 1 \end{array} \right]$ 
C) 
$\left[\begin{array}{rr}
3 & -5 \
-5 & 6 \
-4 & 5
\end{array}\right]$ 
D) 
$\left[\begin{array}{l}
2 \
1 \
2
\end{array}\right]$ 
A) 
$\left[\begin{array}{r}
-2 \
1 \
1
\end{array}\right]$ 
B) 
$\left[ \begin{array} { l l l } - 2 & 1 & 1 \end{array} \right]$ 
C) 
$\left[\begin{array}{rr}
3 & -5 \
-5 & 6 \
-4 & 5
\end{array}\right]$ 
D) 
$\left[\begin{array}{l}
2 \
1 \
2
\end{array}\right]$

Question 5

Solve the matrix equation for X.
-Let $A = \left[ \begin{array} { r r } 5 & - 3 \ 1 & 1 \ 4 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { l l } - 3 & - 7 \ - 3 & - 6 \ - 6 & - 3 \end{array} \right] ; \quad X - B = A$

Accepted Answer

A) 
$\left[ \begin{array} { r r } 2 & - 10 \ - 2 & - 5 \ - 2 & - 2 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } 84 \ 47 \ 102 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 2 & - 10 \ 2 & 1 \ - 2 & 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 2 & 1 \ - 2 & - 5 \ - 2 & - 2 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } 2 & - 10 \ - 2 & - 5 \ - 2 & - 2 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r } 84 \ 47 \ 102 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 2 & - 10 \ 2 & 1 \ - 2 & 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 2 & 1 \ - 2 & - 5 \ - 2 & - 2 \end{array} \right]$

Question 6

Encode and Decode Messages
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual
order.
-Use the coding matrix $\mathrm { A } = \left[ \begin{array} { r r } - 1 & - 3 \ 2 & 5 \end{array} \right]$ to encode the message CARE.

Accepted Answer

A)  $\left[ \begin{array} { r r } - 6 & - 33 \ 11 & 61 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 57 & - 16 \ 96 & 27 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 18 & 105 \ - 7 & - 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 1 & - 8 \ - 4 & - 29 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r } - 6 & - 33 \ 11 & 61 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 57 & - 16 \ 96 & 27 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r } 18 & 105 \ - 7 & - 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 1 & - 8 \ - 4 & - 29 \end{array} \right]$

Question 7

Perform the matrix row operation (or operations)and write the new matrix.
-$\left[ \begin{array} { r r r | r } 36 & 30 & 33 & - 36 \ 1 & 13 & - 3 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right] \frac { 1 } { 3 } \mathrm { R } _ { 1 }$

Accepted Answer

A) 
$\left[ \begin{array} { c c r | r } 12 & 10 & 11 & - 12 \ 1 & 13 & - 3 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r | r } 36 & 30 & 33 & - 36 \ \frac { 1 } { 3 } & \frac { 13 } { 3 } & - 1 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } 12 & 10 & 11 & - 36 \ 1 & 13 & - 3 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { r r r | r } 
12 & 10 & 11 & - 12 \
\frac { 1 } { 3 } & \frac { 13 } { 3 } & - 1 & 0 \
\frac { 2 } { 3 } & - \frac { 7 } { 3 } & \frac { 4 } { 3 } & 7
\end{array} \right]$$ 
A) 
$\left[ \begin{array} { c c r | r } 12 & 10 & 11 & - 12 \ 1 & 13 & - 3 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r | r } 36 & 30 & 33 & - 36 \ \frac { 1 } { 3 } & \frac { 13 } { 3 } & - 1 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } 12 & 10 & 11 & - 36 \ 1 & 13 & - 3 & 0 \ 2 & - 7 & 4 & 21 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { r r r | r } 
12 & 10 & 11 & - 12 \
\frac { 1 } { 3 } & \frac { 13 } { 3 } & - 1 & 0 \
\frac { 2 } { 3 } & - \frac { 7 } { 3 } & \frac { 4 } { 3 } & 7
\end{array} \right]$$

Question 8

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
-$$\begin{array} { r } 
- 4 x - y - 3 z = - 39 \
7 x + 8 y - 4 z = 35 \
3 x - 2 y + z = - 4
\end{array}$$

Accepted Answer

A)  $\{ ( 1,8,9 ) \}$ 
B)  $\{ ( 1,9,8 ) \}$ 
C)  $\{ ( - 1,8,2 ) \}$ 
D)  $\{ ( 2,8 , - 1 ) \}$ 
A)  $\{ ( 1,8,9 ) \}$ 
B)  $\{ ( 1,9,8 ) \}$ 
C)  $\{ ( - 1,8,2 ) \}$ 
D)  $\{ ( 2,8 , - 1 ) \}$

Question 9

Apply Gaussian Elimination to Systems Without Unique Solutions
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{aligned}
x + y + z & = 9 \
2 x - 3 y + 4 z & = 7 \
x - 4 y + 3 z & = - 2
\end{aligned}$$

Accepted Answer

A)  $\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}$ 
B)  $\left\{ \left( \frac { z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}$ 
C)  $\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}$ 
D)  $\left\{ \left( \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}$ 
A)  $\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}$ 
B)  $\left\{ \left( \frac { z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } + \frac { 11 } { 5 } , z \right) \right\}$ 
C)  $\left\{ \left( - \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}$ 
D)  $\left\{ \left( \frac { 7 z } { 5 } + \frac { 34 } { 5 } , \frac { 2 z } { 5 } - \frac { 11 } { 5 } , z \right) \right\}$

Question 10

Use Inverses to Solve Matrix Equations
Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the
constant matrix.
-$\left[ \begin{array} { r r } 7 & 3 \ 4 & 13 \end{array} \right] \left[ \begin{array} { l } x \ y \end{array} \right] = \left[ \begin{array} { l } 11 \ 19 \end{array} \right]$

Accepted Answer

A)  $7 x + 3 y = 11$
$4 x + 13 y = 19$ 
B)  $7 x + 3 y = - 11$
$4 x + 13 y = - 19$ 
C)  $3 x + 7 y = 11$
$4 x + 13 y = 19$ 
D)  $7 x + 3 y = 11$
$13 x + 4 y = 19$ 
A)  $7 x + 3 y = 11$
$4 x + 13 y = 19$ 
B)  $7 x + 3 y = - 11$
$4 x + 13 y = - 19$ 
C)  $3 x + 7 y = 11$
$4 x + 13 y = 19$ 
D)  $7 x + 3 y = 11$
$13 x + 4 y = 19$

Question 11

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
-$$A = \left[ \begin{array} { r r } 
1 & 2 \
0 & - 6
\end{array} \right]$$

Accepted Answer

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

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

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

Question 12

Understand What is Meant by Equal Matrices
Find values for the variables so that the matrices are equal.
-$$\left[ \begin{array} { r r } 
x & y + 5 \
7 z & 10
\end{array} \right] = \left[ \begin{array} { c c } 
3 & 11 \
56 & 10
\end{array} \right]$$

Accepted Answer

A)  $x = 3 ; y = 6 ; z = 8$ 
B)  $x = 3 ; y = 11 ; z = 56$ 
C)  $x = 11 ; y = 10 ; z = 3$ 
D)  $x = 10 ; y = 16 ; z = 392$ 
A)  $x = 3 ; y = 6 ; z = 8$ 
B)  $x = 3 ; y = 11 ; z = 56$ 
C)  $x = 11 ; y = 10 ; z = 3$ 
D)  $x = 10 ; y = 16 ; z = 392$

Question 13

Determinants are used to show that three points lie on the same line (are collinear). If
-The equation of a line passing through two distinct points (x1, y1)and (x2, y2)is given by $\left|\begin{array}{lll}
x & y & 1 \
x_{2} & y 2 & 1 \
x_{3} & y_{3} & 1
\end{array}\right|=0 . \text { Use the determinant to write an equation for the line passing through }(-6,9) \text { and }$
$( - 8,3 )$. Express the line's equation in standard form.

Accepted Answer

A)  $6 x - 2 y + 54 = 0$ 
B)  $9 x - 8 y - 18 = 0$ 
C)  $- 6 x + 2 y + 54 = 0$ 
D)  $3 x - 6 y - 72 = 0$ 
A)  $6 x - 2 y + 54 = 0$ 
B)  $9 x - 8 y - 18 = 0$ 
C)  $- 6 x + 2 y + 54 = 0$ 
D)  $3 x - 6 y - 72 = 0$

Question 14

Solve the problem.
-Let $\mathrm { A } = \left[ \begin{array} { l l } 3 & 3 \ 2 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \ - 1 & 6 \end{array} \right]$. Find $2 \mathrm {~A} + \mathrm { B }$.

Accepted Answer

A) 
$\left[ \begin{array} { l l } 6 & 10 \ 3 & 16 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 6 & 14 \ 2 & 22 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 6 & 10 \ 1 & 11 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 6 & 7 \ 3 & 11 \end{array} \right]$ 
A) 
$\left[ \begin{array} { l l } 6 & 10 \ 3 & 16 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 6 & 14 \ 2 & 22 \end{array} \right]$ 
C) 
$\left[ \begin{array} { l l } 6 & 10 \ 1 & 11 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 6 & 7 \ 3 & 11 \end{array} \right]$

Question 15

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back-substitution to find the solution.
-$\left[ \begin{array} { r r r | r } 1 & \frac { 5 } { 2 } & 1 & \frac { 9 } { 2 } \ 0 & 1 & \frac { 3 } { 2 } & 8 \ 0 & 0 & 1 & - 4 \end{array} \right]$

Accepted Answer

A)  $\left\{ \left( - \frac { 53 } { 2 } , 14 , - 4 \right) \right\}$ 
B)  $\left\{ \left( \frac { 9 } { 2 } , 8 , - 4 \right) \right\}$ 
C)  $\left\{ \left( 0 , \frac { 11 } { 2 } , - 5 \right) \right\}$ 
D)  $\left\{ \left( \frac { 11 } { 2 } , 14 , - 4 \right) \right\}$ 
A)  $\left\{ \left( - \frac { 53 } { 2 } , 14 , - 4 \right) \right\}$ 
B)  $\left\{ \left( \frac { 9 } { 2 } , 8 , - 4 \right) \right\}$ 
C)  $\left\{ \left( 0 , \frac { 11 } { 2 } , - 5 \right) \right\}$ 
D)  $\left\{ \left( \frac { 11 } { 2 } , 14 , - 4 \right) \right\}$

Question 16

Find the product AB, if possible.
-$A = \left[ \begin{array} { l l l } - 9 & - 4 & - 4 \ - 4 & - 9 & - 1 \end{array} \right] , B = \left[ \begin{array} { l } - 1 \ - 3 \ - 9 \end{array} \right]$

Accepted Answer

A) 
$\left[ \begin{array} { l } 57 \ 40 \end{array} \right]$ 
B)  $\mathrm { AB }$ is not defined. 
C)  $\left[ \begin{array} { l l } 57 & 40 \end{array} \right]$ 
D)

$\left[ \begin{array} { c c c } - 9 & - 4 & - 4 \ - 4 & - 9 & - 1 \ - 1 & - 3 & - 9 \end{array} \right]$ 
A) 
$\left[ \begin{array} { l } 57 \ 40 \end{array} \right]$ 
B)  $\mathrm { AB }$ is not defined. 
C)  $\left[ \begin{array} { l l } 57 & 40 \end{array} \right]$ 
D)

$\left[ \begin{array} { c c c } - 9 & - 4 & - 4 \ - 4 & - 9 & - 1 \ - 1 & - 3 & - 9 \end{array} \right]$

Question 17

Model Applied Situations with Matrix Operations
 The $\perp$ shape in the figure below is shown using 9 pixels in a $3 \times 3$ grid. The color levels are given to the right of the figure. Use the matrix $\left[ \begin{array} { l l l } 1 & 3 & 1 \ 1 & 3 & 1 \ 3 & 3 & 3 \end{array} \right]$ that represents a digital photograph of the $\perp$ shape to solve the problem.  
-Using the same color levels from the instructions, write a 3 × 3 matrix A that represents the letter L in dark grey on a white background. Then find a 3 × 3 matrix B so that A + B lightens only the letter L from dark
Grey to light grey.

Accepted Answer

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

Question 18

Evaluate the determinant.
-$$\left| \begin{array} { r r r } 
- 4 & 4 & - 4 \
- 4 & 0 & - 1 \
3 & 0 & 5
\end{array} \right|$$

Accepted Answer

A)  68 
B)  $- 92$ 
C)  $- 68$ 
D)  92 
A)  68 
B)  $- 92$ 
C)  $- 68$ 
D)  92

Question 19

Give the order of the matrix, and identify the given element of the matrix.
-$$\left[ \begin{array} { c c c c c } 
0 & - 15 & - 7 & 13 & 7 \
5 & - e & - 5 & 14 & \pi \
- 6 & 7 & 2 & 1 & - 7 \
\frac { 1 } { 3 } & - 6 & - 8 & 2 & 15
\end{array} \right] ; a _ { 34 }$$

Accepted Answer

A)  $4 \times 5 ; 1$ 
B)  $5 \times 4 ; - 8$ 
C)  $20 ; - 7$ 
D)  $4 \times 4 ; 2$ 
A)  $4 \times 5 ; 1$ 
B)  $5 \times 4 ; - 8$ 
C)  $20 ; - 7$ 
D)  $4 \times 4 ; 2$

Question 20

Model Applied Situations with Matrix Operations
 The $\perp$ shape in the figure below is shown using 9 pixels in a $3 \times 3$ grid. The color levels are given to the right of the figure. Use the matrix $\left[ \begin{array} { l l l } 1 & 3 & 1 \ 1 & 3 & 1 \ 3 & 3 & 3 \end{array} \right]$ that represents a digital photograph of the $\perp$ shape to solve the problem.  
-Adjust the contrast by changing the black to white and the light grey to dark grey. Use matrix addition to accomplish this.

Accepted Answer

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

Use Cramer's rule to solve the system. - 6x-7y-z =-53 x-8y+9z =19 -3x+y+z =11

Use Cramer's rule to solve the system. - 2x+5y=43 2x+2y=22

Solve the problem. -Let $\mathrm { A } = \left[ \begin{array} { r r r } 7 & 2 & 4 \\ 14 & 0 & - 1 \\ 2 & 5 & 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } 5 & - 2 & 1 \\ 2 & 0 & 1 \\ - 3 & 6 & - 5 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

Solve the problem. -Let $A = \left[ \begin{array} { r } 3 \\ - 5 \\ - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r } - 5 \\ 6 \\ 5 \end{array} \right]$ . Find $A + B$ .

Solve the matrix equation for X. -Let $A = \left[ \begin{array} { r r } 5 & - 3 \\ 1 & 1 \\ 4 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { l l } - 3 & - 7 \\ - 3 & - 6 \\ - 6 & - 3 \end{array} \right] ; \quad X - B = A$

Encode and Decode Messages Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order. -Use the coding matrix $\mathrm { A } = \left[ \begin{array} { r r } - 1 & - 3 \\ 2 & 5 \end{array} \right]$ to encode the message CARE.

Perform the matrix row operation (or operations)and write the new matrix. - $\left[ \begin{array} { r r r | r } 36 & 30 & 33 & - 36 \\ 1 & 13 & - 3 & 0 \\ 2 & - 7 & 4 & 21 \end{array} \right] \frac { 1 } { 3 } \mathrm { R } _ { 1 }$

Use Matrices and Gauss-Jordan Elimination to Solve Systems Solve the system of equations using matrices. Use Gauss-Jordan elimination. - -4x-y-3z=-39 7x+8y-4z=35 3x-2y+z=-4

Apply Gaussian Elimination to Systems Without Unique Solutions Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. - x+y+z =9 2x-3y+4z =7 x-4y+3z =-2

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r } 1 & 2 \\0 & - 6\end{array} \right]$

Understand What is Meant by Equal Matrices Find values for the variables so that the matrices are equal. - $\left[ \begin{array} { r r } x & y + 5 \\7 z & 10\end{array} \right] = \left[ \begin{array} { c c } 3 & 11 \\56 & 10\end{array} \right]$

Solve the problem. -Let $\mathrm { A } = \left[ \begin{array} { l l } 3 & 3 \\ 2 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right]$ . Find $2 \mathrm {~A} + \mathrm { B }$ .

Find the product AB, if possible. - $A = \left[ \begin{array} { l l l } - 9 & - 4 & - 4 \\ - 4 & - 9 & - 1 \end{array} \right] , B = \left[ \begin{array} { l } - 1 \\ - 3 \\ - 9 \end{array} \right]$

Evaluate the determinant. - $\left| \begin{array} { r r r } - 4 & 4 & - 4 \\- 4 & 0 & - 1 \\3 & 0 & 5\end{array} \right|$

Give the order of the matrix, and identify the given element of the matrix. - $\left[ \begin{array} { c c c c c } 0 & - 15 & - 7 & 13 & 7 \\5 & - e & - 5 & 14 & \pi \\- 6 & 7 & 2 & 1 & - 7 \\\frac { 1 } { 3 } & - 6 & - 8 & 2 & 15\end{array} \right] ; a _ { 34 }$

Equations and Inequalities

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Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Conic Sections

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Exam 6: Matrices and Determinants