Question 1

Solve a System of Linear Equations in Two Variables Using Cramer's Rule
-$$\begin{array} { l } 
2 x + 3 y = 24 \
2 x - 3 y = 12
\end{array}$$

Accepted Answer

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

Question 2

Solve Matrix Equations
-Let $A = \left[ \begin{array} { r r } 5 & - 2 \ - 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 1 & - 5 \ 5 & 5 \end{array} \right] ; \quad X + A = B$

Accepted Answer

A) 
$X = \left[ \begin{array} { r r } - 6 & - 3 \ 6 & 3 \end{array} \right]$ 
B)

$X = \left[ \begin{array} { r r } - 3 & - 6 \ 3 & 6 \end{array} \right]$ 
C) 
$X = \left[ \begin{array} { r r } 6 & 3 \ - 6 & - 3 \end{array} \right]$ 
D) 
$X = \left[ \begin{array} { r r } 3 & - 6 \ - 3 & 6 \end{array} \right]$ 
A) 
$X = \left[ \begin{array} { r r } - 6 & - 3 \ 6 & 3 \end{array} \right]$ 
B)

$X = \left[ \begin{array} { r r } - 3 & - 6 \ 3 & 6 \end{array} \right]$ 
C) 
$X = \left[ \begin{array} { r r } 6 & 3 \ - 6 & - 3 \end{array} \right]$ 
D) 
$X = \left[ \begin{array} { r r } 3 & - 6 \ - 3 & 6 \end{array} \right]$

Question 3

Multiply Matrices

-$A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \ 5 & 6 & 2 \ - 5 & - 9 & - 8 \end{array} \right]$

Accepted Answer

A)  $[ - 71 - 94 - 67 ]$ 
B) 
$\left[ \begin{array} { l } - 71 \ - 94 \ - 67 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } - 1 & - 9 & 5 \ 1 & - 5 & 9 \ 5 & 6 & 2 \ - 5 & - 9 & - 8 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } - 1 & 45 & 45 \ - 5 & - 54 & 10 \ 5 & 81 & - 40 \end{array} \right]$ 
A)  $[ - 71 - 94 - 67 ]$ 
B) 
$\left[ \begin{array} { l } - 71 \ - 94 \ - 67 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } - 1 & - 9 & 5 \ 1 & - 5 & 9 \ 5 & 6 & 2 \ - 5 & - 9 & - 8 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } - 1 & 45 & 45 \ - 5 & - 54 & 10 \ 5 & 81 & - 40 \end{array} \right]$

Question 4

Perform Scalar Multiplication
-Let $A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$. Find $2 A + 3 B$.

Accepted Answer

A)  $[ - 34 ]$ 
B)  $[ - 64 ]$ 
C)  $\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 0 & 2 \end{array} \right]$ 
A)  $[ - 34 ]$ 
B)  $[ - 64 ]$ 
C)  $\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } 0 & 2 \end{array} \right]$

Question 5

Encode and Decode Messages
-Use the coding matrix $A = \left[ \begin{array} { l l } 2 & 1 \ 5 & 3 \end{array} \right]$ and its inverse $A ^ { - 1 } = \left[ \begin{array} { r r } 3 & - 1 \ - 5 & 2 \end{array} \right]$ to decode the cryptogram $\left[ \begin{array} { r r } 9 & 6 \ 25 & 17 \end{array} \right]$.

Accepted Answer

A)  BEAD 
B)  CARE 
C)  DARE 
D)  CURB 
A)  BEAD 
B)  CARE 
C)  DARE 
D)  CURB

Question 6

Multiply Matrices

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

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
3 & - 7 & 1 \
6 & - 20 & 17
\end{array} \right]$$ 
B)  $\mathrm { AB }$ is not defined. 
C) 
$\left[ \begin{array} { r r } 3 & 6 \ - 7 & - 20 \ 1 & 17 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c c c } 0 & - 6 & 15 \ 1 & - 18 & 12 \end{array} \right]$ 
A) 
$$\left[ \begin{array} { r r r } 
3 & - 7 & 1 \
6 & - 20 & 17
\end{array} \right]$$ 
B)  $\mathrm { AB }$ is not defined. 
C) 
$\left[ \begin{array} { r r } 3 & 6 \ - 7 & - 20 \ 1 & 17 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c c c } 0 & - 6 & 15 \ 1 & - 18 & 12 \end{array} \right]$

Question 7

Multiply Matrices

-$A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \ 0 \ - 3 \end{array} \right]$

Accepted Answer

A)  $[ - 66 ]$ 
B)  $[ 330 ]$ 
C)  $[ - 420 - 24 ]$ 
D) 
$$\left[ \begin{array} { r } 
- 42 \
0 \
- 24
\end{array} \right]$$ 
A)  $[ - 66 ]$ 
B)  $[ 330 ]$ 
C)  $[ - 420 - 24 ]$ 
D) 
$$\left[ \begin{array} { r } 
- 42 \
0 \
- 24
\end{array} \right]$$

Question 8

Solve Problems Involving Systems Without Unique Solutions
-The figure below shows the intersection of three one-way streets. To keep traffic moving, the number of cars per minute entering an intersection must equal the number of cars leaving that intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination to solve the system. If construction limits z to t cars per minute, how many cars per minute must pass through the other intersections to keep traffic moving?

Accepted Answer

A)  $t + 8$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 3$ cars/min between $I _ { 1 }$ and $I _ { 3 }$ 
B)  $t + 1$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 4$ cars/min between $I _ { 1 }$ and $I _ { 3 }$ 
C)  $t - 2$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 1$ cars/min between $I _ { 1 }$ and $I _ { 3 }$ 
D)  $t + 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } - 3$ cars/min between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
A)  $t + 8$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 3$ cars/min between $I _ { 1 }$ and $I _ { 3 }$ 
B)  $t + 1$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 4$ cars/min between $I _ { 1 }$ and $I _ { 3 }$ 
C)  $t - 2$ cars/min between $I _ { 2 }$ and $I _ { 1 } ; t + 1$ cars/min between $I _ { 1 }$ and $I _ { 3 }$ 
D)  $t + 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } - 3$ cars/min between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$

Question 9

Solve a System of Linear Equations in Two Variables Using Cramer's Rule
-$$2 x + 6 y = 26$$
$2 \mathrm { x } + \mathrm { y } = - 4$

Accepted Answer

A)  $\{ ( - 5,6 ) \}$ 
B)  $\{ ( 6 , - 5 ) \}$ 
C)  $\{ ( - 6 , - 5 ) \}$ 
D)  $\{ ( - 5 , - 6 ) \}$ 
A)  $\{ ( - 5,6 ) \}$ 
B)  $\{ ( 6 , - 5 ) \}$ 
C)  $\{ ( - 6 , - 5 ) \}$ 
D)  $\{ ( - 5 , - 6 ) \}$

Question 10

Apply Gaussian Elimination to Systems with More Variables than Equations
-$$\begin{array} { l } 
5 x - y + z = 8 \
7 x + y + z = 6
\end{array}$$

Accepted Answer

A)  $\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}$ 
B)  $\{ ( - z + 3,4 z + 7 , z ) \}$ 
C)  $\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}$ 
B)  $\{ ( - z + 3,4 z + 7 , z ) \}$ 
C)  $\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}$ 
D)  $\varnothing$

Question 11

Perform Matrix Row Operations
-$\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \ - 5 & 0 & 2 & - 3 \ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }$

Accepted Answer

A) 
$\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \ - 30 & 20 & - 3 & - 23 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { c r r | c } 5 & - 4 & 1 & 4 \ 20 & - 20 & 7 & 17 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } 30 & - 4 & - 9 & 19 \ - 5 & 0 & 2 & - 3 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c r r | r } - 30 & 20 & - 3 & - 23 \ - 5 & 0 & 2 & - 3 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \ - 30 & 20 & - 3 & - 23 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { c r r | c } 5 & - 4 & 1 & 4 \ 20 & - 20 & 7 & 17 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r | r } 30 & - 4 & - 9 & 19 \ - 5 & 0 & 2 & - 3 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c r r | r } - 30 & 20 & - 3 & - 23 \ - 5 & 0 & 2 & - 3 \ - 1 & 1 & - 2 & - 1 \end{array} \right]$

Question 12

Multiply Matrices

-$$A = \left[ \begin{array} { r r r } 
- 6 & 5 & - 3 \
- 7 & - 8 & - 5 \
- 2 & 2 & - 7
\end{array} \right] , B = \left[ \begin{array} { r r r } 
1 & - 8 & - 1 \
5 & - 6 & 2 \
- 2 & 1 & 1
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
25 & 15 & 13 \
- 37 & 99 & - 14 \
22 & - 3 & - 1
\end{array} \right]$$ 
B) 
$\left[\begin{array}{rrr}
25 & -37 & 22 \
15 & 99 & -3 \
13 & -14 & -1
\end{array}\right]$ 
C)

$$\left[ \begin{array} { r r r } 
- 6 & 5 & - 3 \
- 7 & - 8 & - 5 \
- 2 & 2 & - 7 \
1 & - 8 & - 1 \
5 & - 6 & 2 \
- 2 & 1 & 1
\end{array} \right]$$ 
D) 
$\left[\begin{array}{rrr}
-6 & -40 & 3 \
-35 & 48 & -10 \
4 & 2 & -7
\end{array}\right]$ 
A) 
$$\left[ \begin{array} { r r r } 
25 & 15 & 13 \
- 37 & 99 & - 14 \
22 & - 3 & - 1
\end{array} \right]$$ 
B) 
$\left[\begin{array}{rrr}
25 & -37 & 22 \
15 & 99 & -3 \
13 & -14 & -1
\end{array}\right]$ 
C)

$$\left[ \begin{array} { r r r } 
- 6 & 5 & - 3 \
- 7 & - 8 & - 5 \
- 2 & 2 & - 7 \
1 & - 8 & - 1 \
5 & - 6 & 2 \
- 2 & 1 & 1
\end{array} \right]$$ 
D) 
$\left[\begin{array}{rrr}
-6 & -40 & 3 \
-35 & 48 & -10 \
4 & 2 & -7
\end{array}\right]$

Question 13

Encode and Decode Messages
-Use the coding matrix $A = \left[ \begin{array} { r r } 1 & - 4 \ - 2 & 9 \end{array} \right]$ and its inverse $A ^ { - 1 } = \left[ \begin{array} { l l } 9 & 4 \ 2 & 1 \end{array} \right]$ to decode the cryptogram $\left[ \begin{array} { r r } - 7 & - 8 \ 16 & 21 \end{array} \right]$.

Accepted Answer

A)  ABLE 
B)  ACTS 
C)  ARMS 
D)  ALAS 
A)  ABLE 
B)  ACTS 
C)  ARMS 
D)  ALAS

Question 14

Write a system of linear equations in three variables, and then use matrices to solve the system.
-The table below shows the number of birds for three selected years after an endangered species protection program was started. $$\begin{array} { l | c c c } 
\mathrm { x } \text { (Number of years after 1980) } & 1 & 5 & 10 \
\hline \mathrm { y } \text { (Number of birds) } & 43 & 139 & 349
\end{array}$$ Use the quadratic function $y = a x ^ { 2 } + b x + c$ to model the data. Solve the system of linear equations involving $a , b$, and $c$ using matrices. Find the equation that models the data.

Accepted Answer

A)  $y = 2 x ^ { 2 } + 12 x + 29$ 
B)  $y = 3 x ^ { 2 } + 24 x + 24$ 
C)  $y = 4 x ^ { 2 } - 12 x + 32$ 
D)  $y = 4 x ^ { 2 } - 36 x + 25$ 
A)  $y = 2 x ^ { 2 } + 12 x + 29$ 
B)  $y = 3 x ^ { 2 } + 24 x + 24$ 
C)  $y = 4 x ^ { 2 } - 12 x + 32$ 
D)  $y = 4 x ^ { 2 } - 36 x + 25$

Question 15

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { l l } 
- 2 & - 6 \
- 6 & - 1
\end{array} \right]$$

Accepted Answer

A)  No inverse 
B) 
$\left[ \begin{array} { r r } 2 & - 6 \ - 6 & - 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } - 2 & 6 \ - 6 & - 1 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { c c } 
- \frac { 1 } { 2 } & - \frac { 1 } { 6 } \
- \frac { 1 } { 6 } & 1
\end{array} \right]$$ 
A)  No inverse 
B) 
$\left[ \begin{array} { r r } 2 & - 6 \ - 6 & - 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } - 2 & 6 \ - 6 & - 1 \end{array} \right]$ 
D) 
$$\left[ \begin{array} { c c } 
- \frac { 1 } { 2 } & - \frac { 1 } { 6 } \
- \frac { 1 } { 6 } & 1
\end{array} \right]$$

Question 16

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$$A = \left[ \begin{array} { l l } 
5 & 3 \
3 & 2
\end{array} \right] , \quad B = \left[ \begin{array} { r r } 
2 & - 3 \
- 3 & 5
\end{array} \right]$$

Accepted Answer

A)  $B = A ^ { - 1 }$ 
B)  $B \neq A ^ { - 1 }$ 
A)  $B = A ^ { - 1 }$ 
B)  $B \neq A ^ { - 1 }$

Question 17

Solve the problem using matrices.
-State University has a College of Arts & Sciences, a College of Business, and a College of Engineering. The percentage of students in each category are given by the following matrix.   The student population is distributed by class and age as given in the following matrix. $$\left. \begin{array} { l c c } 
& \text { Female } & \text { Male } \
\text { Freshman } & 410 & 720 \
\text { Sophomore } & 550 & 750 \
\text { Junior } & 800 & 670 \
\text { Senior } & 630 & 480
\end{array} \right]$$ How many female students are in the College of Business? How many male students are in the College of Arts & Sciences?

Accepted Answer

A)  605 students; 1411 students 
B)  503 students; 693 students 
C)  693 students; 516 students 
D)  1282 students; 605 students 
A)  605 students; 1411 students 
B)  503 students; 693 students 
C)  693 students; 516 students 
D)  1282 students; 605 students

Question 18

Simplify Complex Rational Expressions
-$$\begin{array} { l } 
2 x + 6 y + 9 z = 66 \
7 x + 6 y + 8 z = 77 \
9 x + 8 y - 2 z = 51
\end{array}$$

Accepted Answer

A) 
$\left[ \begin{array} { r r r | r } 2 & 6 & 9 & 66 \ 7 & 6 & 8 & 77 \ 9 & 8 & - 2 & 51 \end{array} \right]$ 
B)

$\left[ \begin{array} { r r r | r } 2 & 7 & 9 & 66 \ 6 & 6 & 8 & 77 \ 9 & 8 & - 2 & 51 \end{array} \right]$ 
C) 
$\left[\begin{array}{rrr|r}
66 & 9 & 6 & 2 \
77 & 8 & 6 & 7 \
51 & -2 & 8 & 9
\end{array}\right]$ 
D) 
$\left[\begin{array}{rrr}
2 & 6 & 9 \
7 & 6 & 8 \
9 & 8 & -2
\end{array}\right]$ 
A) 
$\left[ \begin{array} { r r r | r } 2 & 6 & 9 & 66 \ 7 & 6 & 8 & 77 \ 9 & 8 & - 2 & 51 \end{array} \right]$ 
B)

$\left[ \begin{array} { r r r | r } 2 & 7 & 9 & 66 \ 6 & 6 & 8 & 77 \ 9 & 8 & - 2 & 51 \end{array} \right]$ 
C) 
$\left[\begin{array}{rrr|r}
66 & 9 & 6 & 2 \
77 & 8 & 6 & 7 \
51 & -2 & 8 & 9
\end{array}\right]$ 
D) 
$\left[\begin{array}{rrr}
2 & 6 & 9 \
7 & 6 & 8 \
9 & 8 & -2
\end{array}\right]$

Question 19

Multiplicative Inverses of Matrices and Matrix Equations
1 Find the Multiplicative Inverse of a Square Matrix
-$A = \left[ \begin{array} { r r } - 2 & 4 \ 4 & - 4 \end{array} \right] , \quad B = \left[ \begin{array} { l } \frac { 1 } { 2 } \frac { 1 } { 4 } \ \frac { 1 } { 2 } \frac { 1 } { 4 } \end{array} \right]$

Accepted Answer

A)  $\mathrm { B } \neq \mathrm { A } ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$ 
A)  $\mathrm { B } \neq \mathrm { A } ^ { - 1 }$ 
B)  $B = A ^ { - 1 }$

Question 20

Use Inverses to Solve Matrix Equations
-$$\begin{array} { r } 
5 x + 7 z = 83 \
9 y + 6 z = 72 \
4 x + 7 y - 2 z = 12
\end{array}$$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 5 & 0 & 7 \ 0 & 9 & 6 \ 4 & 7 & - 2 \end{array} \right] \left[ \begin{array} { l } x \ y \ z \end{array} \right] = \left[ \begin{array} { l } 83 \ 72 \ 12 \end{array} \right]$ 
B) 
$$\left[ \begin{array} { r r r } 
5 & 0 & 4 \
0 & 9 & 7 \
7 & 6 & - 2
\end{array} \right] \left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
83 \
72 \
12
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
5 & 7 & 0 \
9 & 6 & 0 \
4 & 7 & - 2
\end{array} \right] \left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
83 \
72 \
12
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { l l } 
5 & 0 \
0 & 9 \
4 & 7
\end{array} \right] \left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
7 \
6 \
- 2
\end{array} \right]$$ 
A)  $\left[ \begin{array} { r r r } 5 & 0 & 7 \ 0 & 9 & 6 \ 4 & 7 & - 2 \end{array} \right] \left[ \begin{array} { l } x \ y \ z \end{array} \right] = \left[ \begin{array} { l } 83 \ 72 \ 12 \end{array} \right]$ 
B) 
$$\left[ \begin{array} { r r r } 
5 & 0 & 4 \
0 & 9 & 7 \
7 & 6 & - 2
\end{array} \right] \left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
83 \
72 \
12
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
5 & 7 & 0 \
9 & 6 & 0 \
4 & 7 & - 2
\end{array} \right] \left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { c } 
83 \
72 \
12
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { l l } 
5 & 0 \
0 & 9 \
4 & 7
\end{array} \right] \left[ \begin{array} { l } 
x \
y \
z
\end{array} \right] = \left[ \begin{array} { l } 
7 \
6 \
- 2
\end{array} \right]$$

Solve a System of Linear Equations in Two Variables Using Cramer's Rule - 2x+3y=24 2x-3y=12

Solve Matrix Equations -Let $A = \left[ \begin{array} { r r } 5 & - 2 \\ - 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 1 & - 5 \\ 5 & 5 \end{array} \right] ; \quad X + A = B$

Multiply Matrices - $A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]$

Perform Scalar Multiplication -Let $A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $2 A + 3 B$ .

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

Multiply Matrices - $A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]$

Solve a System of Linear Equations in Two Variables Using Cramer's Rule - $2 x + 6 y = 26$ $2 \mathrm { x } + \mathrm { y } = - 4$

Apply Gaussian Elimination to Systems with More Variables than Equations - 5x-y+z=8 7x+y+z=6

Perform Matrix Row Operations - $\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }$

Multiply Matrices - $A = \left[ \begin{array} { r r r } - 6 & 5 & - 3 \\- 7 & - 8 & - 5 \\- 2 & 2 & - 7\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 8 & - 1 \\5 & - 6 & 2 \\- 2 & 1 & 1\end{array} \right]$

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { l l } - 2 & - 6 \\- 6 & - 1\end{array} \right]$

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { l l } 5 & 3 \\3 & 2\end{array} \right] , \quad B = \left[ \begin{array} { r r } 2 & - 3 \\- 3 & 5\end{array} \right]$

Simplify Complex Rational Expressions - 2x+6y+9z=66 7x+6y+8z=77 9x+8y-2z=51

Use Inverses to Solve Matrix Equations - 5x+7z=83 9y+6z=72 4x+7y-2z=12

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

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Exam 9: Matrices and Determinants

Solve a System of Linear Equations in Two Variables Using Cramer's Rule - 2x+3y=24 2x-3y=12

Solve Matrix Equations -Let A=[5−2−12]A = \left[ \begin{array} { r r } 5 & - 2 \\ - 1 & 2 \end{array} \right]A=[5−1​−22​] and B=[−1−555];X+A=BB = \left[ \begin{array} { r r } - 1 & - 5 \\ 5 & 5 \end{array} \right] ; \quad X + A = BB=[−15​−55​];X+A=B

Multiply Matrices - A=[−1−95],B=[1−59562−5−9−8]A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]A=[−1​−9​5​],B=​15−5​−56−9​92−8​​

Perform Scalar Multiplication -Let A=[−32]A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]A=[−3​2​] and B=[10]B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]B=[1​0​] . Find 2A+3B2 A + 3 B2A+3B .

Multiply Matrices - A=[−1316],B=[0−251−32]A = \left[ \begin{array} { r r } - 1 & 3 \\ 1 & 6 \end{array} \right] , B = \left[ \begin{array} { l l l } 0 & - 2 & 5 \\ 1 & - 3 & 2 \end{array} \right]A=[−11​36​],B=[01​−2−3​52​]

Multiply Matrices - A=[−628],B=[70−3]A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]A=[−6​2​8​],B=​70−3​​

Solve a System of Linear Equations in Two Variables Using Cramer's Rule - 2x+6y=262 x + 6 y = 262x+6y=26 2x+y=−42 \mathrm { x } + \mathrm { y } = - 42x+y=−4

Apply Gaussian Elimination to Systems with More Variables than Equations - 5x-y+z=8 7x+y+z=6

Perform Matrix Row Operations - [5−414−502−3−11−2−1]−5R1+R2\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }​5−5−1​−401​12−2​4−3−1​​−5R1​+R2​

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - A=[−2−6−6−1]A = \left[ \begin{array} { l l } - 2 & - 6 \\- 6 & - 1\end{array} \right]A=[−2−6​−6−1​]

Simplify Complex Rational Expressions - 2x+6y+9z=66 7x+6y+8z=77 9x+8y-2z=51

Use Inverses to Solve Matrix Equations - 5x+7z=83 9y+6z=72 4x+7y-2z=12

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Solve Matrix Equations -Let $A = \left[ \begin{array} { r r } 5 & - 2 \\ - 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } - 1 & - 5 \\ 5 & 5 \end{array} \right] ; \quad X + A = B$

Multiply Matrices - $A = \left[ \begin{array} { l l l } - 1 & - 9 & 5 \end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 5 & 9 \\ 5 & 6 & 2 \\ - 5 & - 9 & - 8 \end{array} \right]$

Perform Scalar Multiplication -Let $A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $2 A + 3 B$ .

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

Multiply Matrices - $A = \left[ \begin{array} { l l l } - 6 & 2 & 8 \end{array} \right] , B = \left[ \begin{array} { r } 7 \\ 0 \\ - 3 \end{array} \right]$

Solve a System of Linear Equations in Two Variables Using Cramer's Rule - $2 x + 6 y = 26$ $2 \mathrm { x } + \mathrm { y } = - 4$

Perform Matrix Row Operations - $\left[ \begin{array} { r r r | r } 5 & - 4 & 1 & 4 \\ - 5 & 0 & 2 & - 3 \\ - 1 & 1 & - 2 & - 1 \end{array} \right] - 5 R _ { 1 } + R _ { 2 }$

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { l l } - 2 & - 6 \\- 6 & - 1\end{array} \right]$