Question 1

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

Accepted Answer

A)  $\left[ \begin{array} { l l l l } 2 & - 8 & - 10 & 6 \end{array} \right]$ 
B)  $\text { }\left[\begin{array}{llll}
2 & 4 & 5 & -3
\end{array}\right]$ 
C)  $\left[ \begin{array} { l l l l } - 2 & 8 & 10 & - 6 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } - 3 & 2 & 3 & - 5 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l l l } 2 & - 8 & - 10 & 6 \end{array} \right]$ 
B)  $\text { }\left[\begin{array}{llll}
2 & 4 & 5 & -3
\end{array}\right]$ 
C)  $\left[ \begin{array} { l l l l } - 2 & 8 & 10 & - 6 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l l l } - 3 & 2 & 3 & - 5 \end{array} \right]$

Question 2

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.
-$\left[ \begin{array} { l l l } 2 & 5 & 7 \ 9 & 0 & 8 \ 4 & 9 & 0 \end{array} \right] \left[ \begin{array} { l } x \ y \ z \end{array} \right] = \left[ \begin{array} { r } - 2 \ 4 \ 2 \end{array} \right]$

Accepted Answer

A)  $2 x + 5 y + 7 z = - 2$
$9 x + 8 z = 4$
$4 x + 9 y = 2$ 
B)  $2 x - 5 y + 7 z = - 2$
$9 x + 8 z = - 4$
$4 x + 9 y = - 2$ 
C)  $2 x + 5 y + 7 z = - 2$
$9 x + 8 z = 4$
$4 x + 9 z = 2$ 
D)  $2 x + 5 y + 7 z = - 2$
$9 x + 8 y = 4$
$4 x + 9 z = 2$ 
A)  $2 x + 5 y + 7 z = - 2$
$9 x + 8 z = 4$
$4 x + 9 y = 2$ 
B)  $2 x - 5 y + 7 z = - 2$
$9 x + 8 z = - 4$
$4 x + 9 y = - 2$ 
C)  $2 x + 5 y + 7 z = - 2$
$9 x + 8 z = 4$
$4 x + 9 z = 2$ 
D)  $2 x + 5 y + 7 z = - 2$
$9 x + 8 y = 4$
$4 x + 9 z = 2$

Question 3

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
-$$A = \left[ \begin{array} { r r } 
- 6 & 0 \
5 & 3
\end{array} \right]$$

Accepted Answer

A) 
$\left[ \begin{array} { c c } - \frac { 1 } { 6 } & 0 \ \frac { 5 } { 18 } & \frac { 1 } { 3 } \end{array} \right]$ 
B) 
$\left[\begin{array}{c}
-\frac{1}{6} 0 \
-\frac{5}{18} \frac{1}{3}
\end{array}\right]$ 
C)  No inverse 
D)

$\left[ \begin{array} { c c } \frac { 1 } { 3 } & 0 \ \frac { 5 } { 18 } & - \frac { 1 } { 6 } \end{array} \right]$ 
A) 
$\left[ \begin{array} { c c } - \frac { 1 } { 6 } & 0 \ \frac { 5 } { 18 } & \frac { 1 } { 3 } \end{array} \right]$ 
B) 
$\left[\begin{array}{c}
-\frac{1}{6} 0 \
-\frac{5}{18} \frac{1}{3}
\end{array}\right]$ 
C)  No inverse 
D)

$\left[ \begin{array} { c c } \frac { 1 } { 3 } & 0 \ \frac { 5 } { 18 } & - \frac { 1 } { 6 } \end{array} \right]$

Question 4

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

Accepted Answer

A) 
$\left[ \begin{array} { r r } - 12 & 20 \ 0 & 8 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } - 12 & 20 \ 0 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r } - 125 \ 02 \end{array} \right]$ 
D) 
$\left[ \begin{array} { l l } 1 & 9 \ 4 & 6 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } - 12 & 20 \ 0 & 8 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } - 12 & 20 \ 0 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r } - 125 \ 02 \end{array} \right]$ 
D) 
$\left[ \begin{array} { l l } 1 & 9 \ 4 & 6 \end{array} \right]$

Question 5

Understand What is Meant by Equal Matrices
Find values for the variables so that the matrices are equal.
-$$\left[ \begin{array} { r r } 
x + 3 & y + 4 \
7 & 6
\end{array} \right] = \left[ \begin{array} { r r } 
5 & - 4 \
7 & z
\end{array} \right]$$

Accepted Answer

A)  $x = 2 ; y = - 8 ; z = 6$ 
B)  $x = - 2 ; y = 8 ; z = - 6$ 
C)  $x = 5 ; y = - 4 ; z = 6$ 
D)  $x = 2 ; y = 6 ; z = 5$ 
A)  $x = 2 ; y = - 8 ; z = 6$ 
B)  $x = - 2 ; y = 8 ; z = - 6$ 
C)  $x = 5 ; y = - 4 ; z = 6$ 
D)  $x = 2 ; y = 6 ; z = 5$

Question 6

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.

How many female students are in the College of Business? How many male students are in the College of Arts \& Sciences?

Accepted Answer

A)  619 students; 1455 students 
B)  517 students; 711 students 
C)  711 students; 534 students 
D)  1314 students; 619 students 
A)  619 students; 1455 students 
B)  517 students; 711 students 
C)  711 students; 534 students 
D)  1314 students; 619 students

Question 7

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 $\mathrm { A } = \left[ \begin{array} { r r r } 1 & 0 & - 2 \ 1 & 2 & 3 \ 1 & 1 & 1 \end{array} \right]$ to encode the message COME_HERE.

Accepted Answer

A)  $\left[ \begin{array} { r r r } - 23 & - 11 & - 5 \ 72 & 29 & 56 \ 31 & 13 & 28 \end{array} \right]$ 
B) $ \left[\begin{array}{rrr}3 & 5 & 5 \ 15 & 0 & 18 \ 13 & 8 & 5\end{array}\right] $ 
C)  $\left[ \begin{array} { r r r } 19 & 27 & - 21 \ - 14 & - 30 & 39 \ 8 & 11 & - 13 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } - 7 & - 21 & 3 \ 28 & 69 & 44 \ 13 & 33 & 26 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } - 23 & - 11 & - 5 \ 72 & 29 & 56 \ 31 & 13 & 28 \end{array} \right]$ 
B) $ \left[\begin{array}{rrr}3 & 5 & 5 \ 15 & 0 & 18 \ 13 & 8 & 5\end{array}\right] $ 
C)  $\left[ \begin{array} { r r r } 19 & 27 & - 21 \ - 14 & - 30 & 39 \ 8 & 11 & - 13 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } - 7 & - 21 & 3 \ 28 & 69 & 44 \ 13 & 33 & 26 \end{array} \right]$

Question 8

Solve Problems Involving Systems Without Unique Solutions
Solve the problem using matrices.
-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 $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 3 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
B)  $t + 1$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 4 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
C)  $t - 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 1$ cars/min between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
D)  $t + 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } - 3 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
A)  $t + 8$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 3 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
B)  $t + 1$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 4 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
C)  $t - 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } + 1$ cars/min between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$ 
D)  $t + 2$ cars $/ \mathrm { min }$ between $\mathrm { I } _ { 2 }$ and $\mathrm { I } _ { 1 } ; \mathrm { t } - 3 \mathrm { cars } / \mathrm { min }$ between $\mathrm { I } _ { 1 }$ and $\mathrm { I } _ { 3 }$

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.
-$$\begin{array} { r } 
x + y + z = 7 \
x - y + 2 z = 7 \
2 x + 3 z = 14
\end{array}$$

Accepted Answer

A)  $\left\{ \left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right) \right\}$ 
B)  $\left\{ \left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right) \right\}$ 
C)  $\left\{ \left( - \frac { 3 z } { 2 } + 7,2 z , z \right) \right\}$ 
D)  $\left\{ \left( - \frac { 3 z } { 2 } - 7,2 z , z \right) \right\}$ 
A)  $\left\{ \left( - \frac { 3 z } { 2 } + 7 , \frac { z } { 2 } , z \right) \right\}$ 
B)  $\left\{ \left( - \frac { 3 z } { 2 } - 7 , \frac { z } { 2 } , z \right) \right\}$ 
C)  $\left\{ \left( - \frac { 3 z } { 2 } + 7,2 z , z \right) \right\}$ 
D)  $\left\{ \left( - \frac { 3 z } { 2 } - 7,2 z , z \right) \right\}$

Question 10

Solve the problem.
-Let $A = \left[ \begin{array} { r r } - 1 & 4 \ 0 & 4 \ 9 & - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 7 & 2 \ 17 & 4 \ 3 & 2 \end{array} \right]$. Find $A - B$.

Accepted Answer

A) 
$\left[ \begin{array} { r r } - 8 & 2 \ - 17 & 0 \ 6 & - 6 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 1 & 5 \ 7 & 8 \ 12 & 0 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 1 & 2 \ 7 & 0 \ 6 & - 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 3 & - 3 \ 7 & 0 \ - 6 & 6 \end{array} \right]$ 
A) 
$\left[ \begin{array} { r r } - 8 & 2 \ - 17 & 0 \ 6 & - 6 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r } 1 & 5 \ 7 & 8 \ 12 & 0 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 1 & 2 \ 7 & 0 \ 6 & - 2 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 3 & - 3 \ 7 & 0 \ - 6 & 6 \end{array} \right]$

Question 11

Solve the problem.
-$$\left| \begin{array} { l l l } 
x _ { 1 } & y _ { 1 } & 1 \
x _ { 2 } & y _ { 2 } & 1 \
x _ { 3 } & y _ { 3 } & 1
\end{array} \right| = 0 ,$$
then the points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$, and $\left( x _ { 3 } , y _ { 3 } \right)$ are collinear. If the determinant does not equal 0 , then the points are not collinear. Are the points $( 5,7 ) , ( 0,1 )$, and $( 20,25 )$ collinear?

Accepted Answer

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

Question 12

Use Cramer's rule to solve the system.
-$$\begin{array} { l } 
2 x + 2 y = 10 \
2 x - 2 y = 6
\end{array}$$

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 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)  $\mathrm { AB }$ is not defined. 
B) 
$\left[ \begin{array} { r r r } 9 & - 6 & 3 \ - 6 & 12 & - 8 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 9 & - 6 \ - 6 & 12 \ 3 & - 8 \end{array} \right]$ 
D) 
$\left[ \begin{array} { l l } 9 & 0 \ 0 & 8 \end{array} \right]$ 
A)  $\mathrm { AB }$ is not defined. 
B) 
$\left[ \begin{array} { r r r } 9 & - 6 & 3 \ - 6 & 12 & - 8 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 9 & - 6 \ - 6 & 12 \ 3 & - 8 \end{array} \right]$ 
D) 
$\left[ \begin{array} { l l } 9 & 0 \ 0 & 8 \end{array} \right]$

Question 15

Find the product AB, if possible.
-$A = \left[ \begin{array} { l l l } - 5 & 2 & 1 \end{array} \right] , B = \left[ \begin{array} { r r r } - 6 & - 8 & 3 \ - 3 & 7 & - 4 \ - 9 & 2 & 5 \end{array} \right]$

Accepted Answer

A)  $[ 1556 - 18 ]$ 
B) 
$\left[ \begin{array} { r } 15 \ 56 \ - 18 \end{array} \right]$ 
C) 
$$\left[ \begin{array} { r r r } 
- 5 & 2 & 1 \
- 6 & - 8 & 3 \
- 3 & 7 & - 4 \
- 9 & 2 & 5
\end{array} \right] \quad 
D) 
 \left[ \begin{array} { r r r } 
30 & - 16 & 3 \
15 & 14 & - 4 \
45 & 4 & 5
\end{array} \right]$$ 
A)  $[ 1556 - 18 ]$ 
B) 
$\left[ \begin{array} { r } 15 \ 56 \ - 18 \end{array} \right]$ 
C) 
$$\left[ \begin{array} { r r r } 
- 5 & 2 & 1 \
- 6 & - 8 & 3 \
- 3 & 7 & - 4 \
- 9 & 2 & 5
\end{array} \right] \quad 
D) 
 \left[ \begin{array} { r r r } 
30 & - 16 & 3 \
15 & 14 & - 4 \
45 & 4 & 5
\end{array} \right]$$

Question 16

Use Cramer's rule to determine if the system is inconsistent system or contains dependent equations.
-$$\begin{array} { c } 
x + z = 1 \
2 x - 2 y = - 2 \
y + z = 4
\end{array}$$

Accepted Answer

A)  system is inconsistent 
B)  system contains dependent equations 
A)  system is inconsistent 
B)  system contains dependent equations

Question 17

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

Accepted Answer

A)  $\mathrm { AB }$ is not defined. 
B) 
$\left[ \begin{array} { r r r } 12 & - 8 & 4 \ - 6 & 16 & - 11 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 12 & - 6 \ - 8 & 16 \ 4 & - 11 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r } 12 & 0 \ 0 & 12 \end{array} \right]$ 
A)  $\mathrm { AB }$ is not defined. 
B) 
$\left[ \begin{array} { r r r } 12 & - 8 & 4 \ - 6 & 16 & - 11 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } 12 & - 6 \ - 8 & 16 \ 4 & - 11 \end{array} \right]$ 
D)

$\left[ \begin{array} { r r } 12 & 0 \ 0 & 12 \end{array} \right]$

Question 18

Determinants are used to show that three points lie on the same line (are collinear). If
-Determinants are used to show that three points lie on the same line (are collinear). If $$\left| \begin{array} { l l l } 
x _ { 1 } & y _ { 1 } & 1 \
x _ { 2 } & y _ { 2 } & 1 \
x _ { 3 } & y _ { 3 } & 1
\end{array} \right| = 0 ,$$
then the points $\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right) , \left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right)$, and $\left( \mathrm { x } _ { 3 } , \mathrm { y } _ { 3 } \right)$ are collinear. If the determinant does not equal 0 , then the points are not collinear Are the points $( 10 - 9 ) , ( 0,5 )$ and $( 30 - 36 )$ collinear?

Accepted Answer

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

Question 19

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{array} { c } 
x + y + z = 9 \
2 x - 3 y + 4 z = 7
\end{array}$$

Accepted Answer

A)  $\left\{ \left( - \frac { 7 } { 5 } z + \frac { 34 } { 5 } , \frac { 2 } { 5 } z + \frac { 11 } { 5 } , z \right) \right\}$ 
B)  $\left\{ \left( \frac { 3 } { 5 } z + \frac { 16 } { 5 } , - \frac { 8 } { 5 } z + \frac { 29 } { 5 } , z \right) \right\}$ 
C)  $\left\{ \left( \frac { 27 } { 5 } , \frac { 13 } { 5 } , 1 \right) \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \left( - \frac { 7 } { 5 } z + \frac { 34 } { 5 } , \frac { 2 } { 5 } z + \frac { 11 } { 5 } , z \right) \right\}$ 
B)  $\left\{ \left( \frac { 3 } { 5 } z + \frac { 16 } { 5 } , - \frac { 8 } { 5 } z + \frac { 29 } { 5 } , z \right) \right\}$ 
C)  $\left\{ \left( \frac { 27 } { 5 } , \frac { 13 } { 5 } , 1 \right) \right\}$ 
D)  $\varnothing$

Question 20

Use Cramer's rule to solve the system.
-$$\begin{array} { l } 
5 x = - 5 y + 10 \
4 x = - y - 10
\end{array}$$

Accepted Answer

A)  $\{ ( - 4,6 ) \}$ 
B)  $\{ ( 6 , - 4 ) \}$ 
C)  $\{ ( - 6 , - 4 ) \}$ 
D)  $\{ ( - 4 , - 6 ) \}$ 
A)  $\{ ( - 4,6 ) \}$ 
B)  $\{ ( 6 , - 4 ) \}$ 
C)  $\{ ( - 6 , - 4 ) \}$ 
D)  $\{ ( - 4 , - 6 ) \}$

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

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r } - 6 & 0 \\5 & 3\end{array} \right]$

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

Understand What is Meant by Equal Matrices Find values for the variables so that the matrices are equal. - $\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 6\end{array} \right] = \left[ \begin{array} { r r } 5 & - 4 \\7 & z\end{array} \right]$

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=7 x-y+2z=7 2x+3z=14

Solve the problem. -Let $A = \left[ \begin{array} { r r } - 1 & 4 \\ 0 & 4 \\ 9 & - 4 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 7 & 2 \\ 17 & 4 \\ 3 & 2 \end{array} \right]$ . Find $A - B$ .

Use Cramer's rule to solve the system. - 2x+2y=10 2x-2y=6

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

Find the product AB, if possible. - $A = \left[ \begin{array} { l l l } - 5 & 2 & 1 \end{array} \right] , B = \left[ \begin{array} { r r r } - 6 & - 8 & 3 \\ - 3 & 7 & - 4 \\ - 9 & 2 & 5 \end{array} \right]$

Use Cramer's rule to determine if the system is inconsistent system or contains dependent equations. - x+z=1 2x-2y=-2 y+z=4

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

Apply Gaussian Elimination to Systems with More Variables than Equations 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

Use Cramer's rule to solve the system. - 5x=-5y+10 4x=-y-10

Equations and Inequalities

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Exam 6: Matrices and Determinants

Solve the problem. -Let B=[−145−3]\mathrm { B } = \left[ \begin{array} { l l l l } - 1 & 4 & 5 & - 3 \end{array} \right]B=[−1​4​5​−3​] . Find −2 B- 2 \mathrm {~B}−2 B .

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - A=[−6053]A = \left[ \begin{array} { r r } - 6 & 0 \\5 & 3\end{array} \right]A=[−65​03​]

Solve the problem. -Let A=[−3502]\mathrm { A } = \left[ \begin{array} { r r } - 3 & 5 \\ 0 & 2 \end{array} \right]A=[−30​52​] . Find 4 A4 \mathrm {~A}4 A .

Understand What is Meant by Equal Matrices Find values for the variables so that the matrices are equal. - [x+3y+476]=[5−47z]\left[ \begin{array} { r r } x + 3 & y + 4 \\7 & 6\end{array} \right] = \left[ \begin{array} { r r } 5 & - 4 \\7 & z\end{array} \right][x+37​y+46​]=[57​−4z​]