Question 1

If $A$ is a $3 \times 4$ matrix and the matrix sum $A+B$ exists, what can you say about the matrix $B$ ?

Accepted Answer

A)  B is a matrix with 3 rows 
B)  B is a $3 \times 4$ matrix 
C)  B is a matrix with 4 rows 
D)  B is a matrix with 4 columns 
A)  B is a matrix with 3 rows 
B)  B is a $3 \times 4$ matrix 
C)  B is a matrix with 4 rows 
D)  B is a matrix with 4 columns

Question 2

Find the inverse, if it exists, of the given matrix.
-$\left[\begin{array}{rrr}-2 & 1 & 3 \ 3 & -1 & 2 \ -4 & 2 & 0\end{array}\right]$

Accepted Answer

A) $$\left[\begin{array}{ccc}
-\frac{2}{3} & 1 & \frac{5}{6} \
-1 & 2 & 2 \
\frac{1}{3} & 0 & -\frac{1}{6}
\end{array}\right]$$ 
B) $$\left[\begin{array}{ccc}
-1 & 1 & \frac{5}{6} \
-\frac{4}{3} & 2 & \frac{13}{6} \
\frac{1}{3} & 0 & -\frac{1}{6}
\end{array}\right]$$ 
C)  $\left[\begin{array}{ccc}-\frac{2}{3} & 1 & \frac{5}{6} \ -\frac{4}{3} & 2 & \frac{13}{6} \ \frac{1}{3} & 0 & 0\end{array}\right]$ 
D) $$\left[\begin{array}{ccc}
-\frac{2}{3} & 1 & \frac{5}{6} \
-\frac{4}{3} & 2 & \frac{13}{6} \
\frac{1}{3} & 0 & -\frac{1}{6}
\end{array}\right]$$ 
A) $$\left[\begin{array}{ccc}
-\frac{2}{3} & 1 & \frac{5}{6} \
-1 & 2 & 2 \
\frac{1}{3} & 0 & -\frac{1}{6}
\end{array}\right]$$ 
B) $$\left[\begin{array}{ccc}
-1 & 1 & \frac{5}{6} \
-\frac{4}{3} & 2 & \frac{13}{6} \
\frac{1}{3} & 0 & -\frac{1}{6}
\end{array}\right]$$ 
C)  $\left[\begin{array}{ccc}-\frac{2}{3} & 1 & \frac{5}{6} \ -\frac{4}{3} & 2 & \frac{13}{6} \ \frac{1}{3} & 0 & 0\end{array}\right]$ 
D) $$\left[\begin{array}{ccc}
-\frac{2}{3} & 1 & \frac{5}{6} \
-\frac{4}{3} & 2 & \frac{13}{6} \
\frac{1}{3} & 0 & -\frac{1}{6}
\end{array}\right]$$

Question 3

Solve the problem.
-The infant mortality rate at three different hospitals is given in the table below.
 
The table below shows the number of live births at each of these hospitals in each of the years 1990-1992.
 
What matrix product displays the total number of infant deaths at the three hospitals in each of the years 1990-1992? Display the two matrices which must be multiplied and their product. Round numbers to the nearest whole number.

Accepted Answer

A)  $\left[\begin{array}{rrr}1850 & 2420 & 970 \ 1218 & 1960 & 708 \ 1176 & 2190 & 1075\end{array}\right]\left[\begin{array}{l}0.007 \ 0.006 \ 0.007\end{array}\right]=\left[\begin{array}{rrr}13 & 157 \ 9 & 12 & 5 \ 8 & 132 & 8\end{array}\right]$ 
B) 
$$\left[\begin{array}{rrr}
1850 & 2420 & 970 \
1218 & 1960 & 708 \
1176 & 2190 & 1075
\end{array}\right]\left[\begin{array}{l}
0)007 \
0)006 \
0)007
\end{array}\right]=\left[\begin{array}{l}
34 \
25 \
29
\end{array}\right]$$ 
C) 
$$\left[\begin{array}{rrr}
0)007 & 0.006 & 0.007
\end{array}\right]\left[\begin{array}{rrr}
1850 & 2420 & 970 \
1218 & 1960 & 708 \
1176 & 2190 & 1075
\end{array}\right]=\left[\begin{array}{l}
28 \
44 \
19
\end{array}\right]$$ 
D) $$\left[\begin{array}{lll}
1850 & 2420 & 970
\end{array}\right]\left[\begin{array}{l}
0)007 \
0)006 \
0)007
\end{array}\right]=[34]$$ 
A)  $\left[\begin{array}{rrr}1850 & 2420 & 970 \ 1218 & 1960 & 708 \ 1176 & 2190 & 1075\end{array}\right]\left[\begin{array}{l}0.007 \ 0.006 \ 0.007\end{array}\right]=\left[\begin{array}{rrr}13 & 157 \ 9 & 12 & 5 \ 8 & 132 & 8\end{array}\right]$ 
B) 
$$\left[\begin{array}{rrr}
1850 & 2420 & 970 \
1218 & 1960 & 708 \
1176 & 2190 & 1075
\end{array}\right]\left[\begin{array}{l}
0)007 \
0)006 \
0)007
\end{array}\right]=\left[\begin{array}{l}
34 \
25 \
29
\end{array}\right]$$ 
C) 
$$\left[\begin{array}{rrr}
0)007 & 0.006 & 0.007
\end{array}\right]\left[\begin{array}{rrr}
1850 & 2420 & 970 \
1218 & 1960 & 708 \
1176 & 2190 & 1075
\end{array}\right]=\left[\begin{array}{l}
28 \
44 \
19
\end{array}\right]$$ 
D) $$\left[\begin{array}{lll}
1850 & 2420 & 970
\end{array}\right]\left[\begin{array}{l}
0)007 \
0)006 \
0)007
\end{array}\right]=[34]$$

Question 4

Given the matrices $A$ and $B$, find the matrix product $A B$.
-$\mathrm{A}=\left[\begin{array}{rrr}3 & -2 & 1 \ 0 & 4 & -3\end{array}\right], \mathrm{B}=\left[\begin{array}{rr}5 & 0 \ -2 & 1\end{array}\right]$ Find $\mathrm{AB}$.

Accepted Answer

A) $$\left[\begin{array}{rrr}
15 & -10 & 5 \
-6 & 8 & -5
\end{array}\right]$$ 
B)  $A B$ is not defined. 
C) 
$$\left[\begin{array}{rr}
15 & -6 \
-10 & 8 \
5 & -5
\end{array}\right]$$ 
D) $\left[\begin{array}{rr}15 & 0 \ 0 & 4\end{array}\right]$ 
A) $$\left[\begin{array}{rrr}
15 & -10 & 5 \
-6 & 8 & -5
\end{array}\right]$$ 
B)  $A B$ is not defined. 
C) 
$$\left[\begin{array}{rr}
15 & -6 \
-10 & 8 \
5 & -5
\end{array}\right]$$ 
D) $\left[\begin{array}{rr}15 & 0 \ 0 & 4\end{array}\right]$

Question 5

Perform the indicated operation where possible.
-$\left[\begin{array}{rr}-7 & 1 \ 2 & 5\end{array}\right]+\left[\begin{array}{rr}6 & 2 \ 2 & -3\end{array}\right]$

Accepted Answer

A) $\left[\begin{array}{rr}3 & 4 \ -2 & 2\end{array}\right]$ 
B) $\left[\begin{array}{rr}-1 & 3 \ 4 & 2\end{array}\right]$ 
C) $\left[\begin{array}{cc}-1 & -5 \ -5 & -10\end{array}\right]$ 
D) $[8]$ 
A) $\left[\begin{array}{rr}3 & 4 \ -2 & 2\end{array}\right]$ 
B) $\left[\begin{array}{rr}-1 & 3 \ 4 & 2\end{array}\right]$ 
C) $\left[\begin{array}{cc}-1 & -5 \ -5 & -10\end{array}\right]$ 
D) $[8]$

Question 6

Determine whether the two matrices are inverses of each other by computing their product.
-$\left[\begin{array}{rr}-5 & -1 \ 6 & 0\end{array}\right],\left[\begin{array}{r}0 \frac{1}{6} \ -1 \frac{5}{6}\end{array}\right]$

Accepted Answer

A) True 
 B)False

Question 7

Use the Gauss-Jordan method to solve the system of equations.
-$x-y+3 z=-8$
$x+5 y+z=40$
$5 x+y+13 z=10$

Accepted Answer

A)  $(8,0,8)$ 
B)  No solution 
C)  $(0,8,0)$ 
D)  $(8,8,0)$ 
A)  $(8,0,8)$ 
B)  No solution 
C)  $(0,8,0)$ 
D)  $(8,8,0)$

Question 8

Write a system of equations and use the inverse of the coefficient matrix to solve the system.
-A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. Cable B requires 1 black, 2 white, and 1 red wires. Cable $C$ requires 2 black, 1 white, and 2 red wires. They used 135 black, 70 white and 80 red wires. How many of each cable were made?

Accepted Answer

A)  5 cable A
18 cable B
25 cable C 
B)  58 cable A
30 cable B
22 cable C 
C)  30 cable A
5 cable B
25 cable C 
D)  15 cable A
20 cable B
15 cable 
A)  5 cable A
18 cable B
25 cable C 
B)  58 cable A
30 cable B
22 cable C 
C)  30 cable A
5 cable B
25 cable C 
D)  15 cable A
20 cable B
15 cable

Question 9

Solve the system of two equations in two variables.
-$7 x+5 y=-5$
$-3 x-2 y=2$

Accepted Answer

A)  $(-1,0)$ 
B)  $(0,-1)$ 
C)  No solution 
D)  $(0,0)$ 
A)  $(-1,0)$ 
B)  $(0,-1)$ 
C)  No solution 
D)  $(0,0)$

Question 10

Find the inverse, if it exists, of the given matrix.
-$\mathrm{A}=\left[\begin{array}{ll}4 & 0 \ 2 & 3\end{array}\right]$

Accepted Answer

A) $\left[\begin{array}{ll}\frac{1}{4} & 0 \ \frac{1}{6} & \frac{1}{3}\end{array}\right]$ 
B) $$\left[\begin{array}{cc}
\frac{1}{4} & 0 \
-\frac{1}{6} & \frac{1}{3}
\end{array}\right]$$ 
C)  No inverse 
D) $$\left[\begin{array}{cc}
\frac{1}{3} & 0 \
-\frac{1}{6} & \frac{1}{4}
\end{array}\right]$$ 
A) $\left[\begin{array}{ll}\frac{1}{4} & 0 \ \frac{1}{6} & \frac{1}{3}\end{array}\right]$ 
B) $$\left[\begin{array}{cc}
\frac{1}{4} & 0 \
-\frac{1}{6} & \frac{1}{3}
\end{array}\right]$$ 
C)  No inverse 
D) $$\left[\begin{array}{cc}
\frac{1}{3} & 0 \
-\frac{1}{6} & \frac{1}{4}
\end{array}\right]$$

Question 11

Find the order of the matrix product $\mathrm{AB}$ and the product $\mathrm{BA}$, whenever the products exist.
-$\mathrm{A}$ is $2 \times 1, \mathrm{~B}$ is $1 \times 1$.

Accepted Answer

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

Question 12

Determine whether the two matrices are inverses of each other by computing their product.
-$\left[\begin{array}{rrr}1 & 2 & 0 \ -2 & 0 & 2 \ 3 & 2 & -4\end{array}\right]$ and $\left[\begin{array}{rrr}\frac{1}{2} & -1 & -\frac{1}{2} \ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2}\end{array}\right]$

Accepted Answer

A) True 
 B)False

Question 13

What is the size of the matrix?

Accepted Answer

A)  $3 \times 0$ 
B)  $3 \times 1$ 
C)  $1 \times 3$ 
D)  $3 \times 3$ 
A)  $3 \times 0$ 
B)  $3 \times 1$ 
C)  $1 \times 3$ 
D)  $3 \times 3$

Question 14

Solve the problem.
-According to an agricultural report, the amounts of nitrogen (lb/acre), phosphate (lb/acre), and labor (lb/acre), needed to grow three different vegetables are given by the following table:
 
If a farmer has 260 acres, 36,200 pounds of nitrogen, 30,000 pounds of phosphate, and 1200 hours of labor, can he use all of his resources completely? If so, how many acres should he allot for each crop?

Accepted Answer

A)  Yes; 120 acres for vegetable A, 100 acres for vegetable B, 40 acres for vegetable C 
B)  No 
C)  Yes; 110 acres for vegetable A, 110 acres for vegetable B, 40 acres for vegetable C 
D)  Yes; 120 acres for vegetable A, 120 acres for vegetable B, 60 acres for vegetable C 
A)  Yes; 120 acres for vegetable A, 100 acres for vegetable B, 40 acres for vegetable C 
B)  No 
C)  Yes; 110 acres for vegetable A, 110 acres for vegetable B, 40 acres for vegetable C 
D)  Yes; 120 acres for vegetable A, 120 acres for vegetable B, 60 acres for vegetable C

Question 15

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$.
-$2 x+y-2 z=12$
$4 \mathrm{x}-4 \mathrm{y}+6 \mathrm{z}=14$

Accepted Answer

A)  $\left(\frac{31}{6}+\frac{1}{6} z, \frac{5}{3}+\frac{5}{3} z, z\right)$ 
B)  $\left(\frac{31}{6}+\frac{1}{6} z, 12-2 x+2 z, z\right)$ 
C)  $\left(\frac{31}{6}+\frac{1}{6} z, 12+2 x-2 z, z\right)$ 
D)  $(62+2 z, 20+62 z, z)$ 
A)  $\left(\frac{31}{6}+\frac{1}{6} z, \frac{5}{3}+\frac{5}{3} z, z\right)$ 
B)  $\left(\frac{31}{6}+\frac{1}{6} z, 12-2 x+2 z, z\right)$ 
C)  $\left(\frac{31}{6}+\frac{1}{6} z, 12+2 x-2 z, z\right)$ 
D)  $(62+2 z, 20+62 z, z)$

Question 16

Perform the indicated operation.
-Let $B=\left[\begin{array}{llll}-1 & 2 & 6 & -3\end{array}\right]$. Find $-4 B$.

Accepted Answer

A)  $\left[\begin{array}{llll}4 & -8 & -24 & 12\end{array}\right]$ 
B)  $\left[\begin{array}{llll}4 & 2 & 6 & -3\end{array}\right]$ 
C)  $\left[\begin{array}{llll}-3 & 0 & 4 & -5\end{array}\right]$ 
D)  $\left[\begin{array}{llll}-4 & 8 & 24 & -12\end{array}\right]$ 
A)  $\left[\begin{array}{llll}4 & -8 & -24 & 12\end{array}\right]$ 
B)  $\left[\begin{array}{llll}4 & 2 & 6 & -3\end{array}\right]$ 
C)  $\left[\begin{array}{llll}-3 & 0 & 4 & -5\end{array}\right]$ 
D)  $\left[\begin{array}{llll}-4 & 8 & 24 & -12\end{array}\right]$

Question 17

Write an augmented matrix for the system of equations.
-$9 x+4 y=5$
$8 x-2 y=10$

Accepted Answer

A) $\left[\begin{array}{rr|r}9 & 4 & 5 \ 8 & -2 & 10\end{array}\right]$ 
B) $\left[\begin{array}{rr|r}5 & 4 & 9 \ 10 & 8 & -2\end{array}\right]$ 
C) $\left[\begin{array}{rr|r}9 & 8 & 5 \ 4 & -2 & 10\end{array}\right]$ 
D) $\left[\begin{array}{rr|r}9 & 4 & 10 \ -2 & 8 & 5\end{array}\right]$ 
A) $\left[\begin{array}{rr|r}9 & 4 & 5 \ 8 & -2 & 10\end{array}\right]$ 
B) $\left[\begin{array}{rr|r}5 & 4 & 9 \ 10 & 8 & -2\end{array}\right]$ 
C) $\left[\begin{array}{rr|r}9 & 8 & 5 \ 4 & -2 & 10\end{array}\right]$ 
D) $\left[\begin{array}{rr|r}9 & 4 & 10 \ -2 & 8 & 5\end{array}\right]$

Question 18

What is the size of the matrix?
$\left[\begin{array}{llll}3 & -8 & 3 & 7\end{array}\right]$

Accepted Answer

A)  $1 \times 4$ 
B)  $4 \times 1$ 
C)  1 
D)  4 
A)  $1 \times 4$ 
B)  $4 \times 1$ 
C)  1 
D)  4

Question 19

Solve the problem by writing and solving a suitable system of equations.
-Linda invests $\$ 25,000$ for one year. Part is invested at $5 \%$, another part at $6 \%$, and the rest at $8 \%$. The total income from all 3 investments is $\$ 1600$. The income from the $5 \%$ and $6 \%$ investments is the same as the income from the $8 \%$ investment. Find the amount invested at each rate.

Accepted Answer

A)  $\$ 10,000$ at $5 \%, \$ 5000$ at $6 \%, \$ 10,000$ at $8 \%$ 
B)  $\$ 10,000$ at $5 \%, \$ 10,000$ at $6 \%, \$ 5000$ at $8 \%$ 
C)  $\$ 8000$ at $5 \%, \$ 10,000$ at $6 \%, \$ 7000$ at $8 \%$ 
D)  $\$ 5000$ at $5 \%, \$ 10,000$ at $6 \%, \$ 10,000$ at $8 \%$ 
A)  $\$ 10,000$ at $5 \%, \$ 5000$ at $6 \%, \$ 10,000$ at $8 \%$ 
B)  $\$ 10,000$ at $5 \%, \$ 10,000$ at $6 \%, \$ 5000$ at $8 \%$ 
C)  $\$ 8000$ at $5 \%, \$ 10,000$ at $6 \%, \$ 7000$ at $8 \%$ 
D)  $\$ 5000$ at $5 \%, \$ 10,000$ at $6 \%, \$ 10,000$ at $8 \%$

Question 20

Determine whether the two matrices are inverses of each other by computing their product.
-$\left[\begin{array}{rrr}2 & -1 & 0 \ -1 & 1 & -2 \ 1 & 0 & -1\end{array}\right],\left[\begin{array}{rrr}1 & -1 & 2 \ -3 & -2 & 4 \ -1 & 1 & 1\end{array}\right]$

Accepted Answer

A) True 
 B)False

If $A$ is a $3 \times 4$ matrix and the matrix sum $A+B$ exists, what can you say about the matrix $B$ ?

Find the inverse, if it exists, of the given matrix. - $\left[\begin{array}{rrr}-2 & 1 & 3 \\ 3 & -1 & 2 \\ -4 & 2 & 0\end{array}\right]$

Given the matrices $A$ and $B$ , find the matrix product $A B$ . - $\mathrm{A}=\left[\begin{array}{rrr}3 & -2 & 1 \\ 0 & 4 & -3\end{array}\right], \mathrm{B}=\left[\begin{array}{rr}5 & 0 \\ -2 & 1\end{array}\right]$ Find $\mathrm{AB}$ .

Perform the indicated operation where possible. - $\left[\begin{array}{rr}-7 & 1 \\ 2 & 5\end{array}\right]+\left[\begin{array}{rr}6 & 2 \\ 2 & -3\end{array}\right]$

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{rr}-5 & -1 \\ 6 & 0\end{array}\right],\left[\begin{array}{r}0 \frac{1}{6} \\ -1 \frac{5}{6}\end{array}\right]$

Use the Gauss-Jordan method to solve the system of equations. - $x-y+3 z=-8$ $x+5 y+z=40$ $5 x+y+13 z=10$

Solve the system of two equations in two variables. - $7 x+5 y=-5$ $-3 x-2 y=2$

Find the inverse, if it exists, of the given matrix. - $\mathrm{A}=\left[\begin{array}{ll}4 & 0 \\ 2 & 3\end{array}\right]$

Find the order of the matrix product $\mathrm{AB}$ and the product $\mathrm{BA}$ , whenever the products exist. - $\mathrm{A}$ is $2 \times 1, \mathrm{~B}$ is $1 \times 1$ .

What is the size of the matrix?

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ . - $2 x+y-2 z=12$ $4 \mathrm{x}-4 \mathrm{y}+6 \mathrm{z}=14$

Perform the indicated operation. -Let $B=\left[\begin{array}{llll}-1 & 2 & 6 & -3\end{array}\right]$ . Find $-4 B$ .

Write an augmented matrix for the system of equations. - $9 x+4 y=5$ $8 x-2 y=10$

What is the size of the matrix? $\left[\begin{array}{llll}3 & -8 & 3 & 7\end{array}\right]$

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{rrr}2 & -1 & 0 \\ -1 & 1 & -2 \\ 1 & 0 & -1\end{array}\right],\left[\begin{array}{rrr}1 & -1 & 2 \\ -3 & -2 & 4 \\ -1 & 1 & 1\end{array}\right]$

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Linear Programming

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Applications of the Derivative

Integral Calculus

Multivariate Calculus

Filters

Exam 6: Systems of Linear Equations and Matrices

If AAA is a 3×43 \times 43×4 matrix and the matrix sum A+BA+BA+B exists, what can you say about the matrix BBB ?

Find the inverse, if it exists, of the given matrix. - [−2133−12−420]\left[\begin{array}{rrr}-2 & 1 & 3 \\ 3 & -1 & 2 \\ -4 & 2 & 0\end{array}\right]​−23−4​1−12​320​​

Perform the indicated operation where possible. - [−7125]+[622−3]\left[\begin{array}{rr}-7 & 1 \\ 2 & 5\end{array}\right]+\left[\begin{array}{rr}6 & 2 \\ 2 & -3\end{array}\right][−72​15​]+[62​2−3​]

Determine whether the two matrices are inverses of each other by computing their product. - [−5−160],[016−156]\left[\begin{array}{rr}-5 & -1 \\ 6 & 0\end{array}\right],\left[\begin{array}{r}0 \frac{1}{6} \\ -1 \frac{5}{6}\end{array}\right][−56​−10​],[061​−165​​]

Use the Gauss-Jordan method to solve the system of equations. - x−y+3z=−8x-y+3 z=-8x−y+3z=−8 x+5y+z=40x+5 y+z=40x+5y+z=40 5x+y+13z=105 x+y+13 z=105x+y+13z=10

Solve the system of two equations in two variables. - 7x+5y=−57 x+5 y=-57x+5y=−5 −3x−2y=2-3 x-2 y=2−3x−2y=2

Find the inverse, if it exists, of the given matrix. - A=[4023]\mathrm{A}=\left[\begin{array}{ll}4 & 0 \\ 2 & 3\end{array}\right]A=[42​03​]

Find the order of the matrix product AB\mathrm{AB}AB and the product BA\mathrm{BA}BA , whenever the products exist. - A\mathrm{A}A is 2×1, B2 \times 1, \mathrm{~B}2×1, B is 1×11 \times 11×1 .

What is the size of the matrix?

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z\mathrm{z}z . - 2x+y−2z=122 x+y-2 z=122x+y−2z=12 4x−4y+6z=144 \mathrm{x}-4 \mathrm{y}+6 \mathrm{z}=144x−4y+6z=14

Perform the indicated operation. -Let B=[−126−3]B=\left[\begin{array}{llll}-1 & 2 & 6 & -3\end{array}\right]B=[−1​2​6​−3​] . Find −4B-4 B−4B .

Write an augmented matrix for the system of equations. - 9x+4y=59 x+4 y=59x+4y=5 8x−2y=108 x-2 y=108x−2y=10

What is the size of the matrix? [3−837]\left[\begin{array}{llll}3 & -8 & 3 & 7\end{array}\right][3​−8​3​7​]

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Linear Programming

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Applications of the Derivative

Integral Calculus

Multivariate Calculus

Filters

If $A$ is a $3 \times 4$ matrix and the matrix sum $A+B$ exists, what can you say about the matrix $B$ ?

Find the inverse, if it exists, of the given matrix. - $\left[\begin{array}{rrr}-2 & 1 & 3 \\ 3 & -1 & 2 \\ -4 & 2 & 0\end{array}\right]$

Perform the indicated operation where possible. - $\left[\begin{array}{rr}-7 & 1 \\ 2 & 5\end{array}\right]+\left[\begin{array}{rr}6 & 2 \\ 2 & -3\end{array}\right]$

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{rr}-5 & -1 \\ 6 & 0\end{array}\right],\left[\begin{array}{r}0 \frac{1}{6} \\ -1 \frac{5}{6}\end{array}\right]$

Use the Gauss-Jordan method to solve the system of equations. - $x-y+3 z=-8$ $x+5 y+z=40$ $5 x+y+13 z=10$

Solve the system of two equations in two variables. - $7 x+5 y=-5$ $-3 x-2 y=2$

Find the inverse, if it exists, of the given matrix. - $\mathrm{A}=\left[\begin{array}{ll}4 & 0 \\ 2 & 3\end{array}\right]$

Find the order of the matrix product $\mathrm{AB}$ and the product $\mathrm{BA}$ , whenever the products exist. - $\mathrm{A}$ is $2 \times 1, \mathrm{~B}$ is $1 \times 1$ .

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ . - $2 x+y-2 z=12$ $4 \mathrm{x}-4 \mathrm{y}+6 \mathrm{z}=14$

Perform the indicated operation. -Let $B=\left[\begin{array}{llll}-1 & 2 & 6 & -3\end{array}\right]$ . Find $-4 B$ .

Write an augmented matrix for the system of equations. - $9 x+4 y=5$ $8 x-2 y=10$

What is the size of the matrix? $\left[\begin{array}{llll}3 & -8 & 3 & 7\end{array}\right]$