Question 1

Find the complete solution of the system. $$\left\{ \begin{array} { r } 
x + y + z = 3 \
x - y + z = - 2 \
x + y + 3 z = 1
\end{array} \right.$$

Accepted Answer

A)  $( 3,5,1 )$ 
B)  $\left( \frac { 3 } { 2 } , \frac { 5 } { 2 } , - 1 \right)$ 
C)  $\left( 0 , \frac { 5 } { 2 } , - 1 \right)$ 
D)  $\left( 2 , \frac { 3 } { 2 } , - 1 \right)$ 
E) none 
A)  $( 3,5,1 )$ 
B)  $\left( \frac { 3 } { 2 } , \frac { 5 } { 2 } , - 1 \right)$ 
C)  $\left( 0 , \frac { 5 } { 2 } , - 1 \right)$ 
D)  $\left( 2 , \frac { 3 } { 2 } , - 1 \right)$ 
E) none B

Question 2

Find inverse of the matrix. $\left[ \begin{array} { l l l } 
2 & 2 & 0 \
0 & 2 & 0 \
1 & 0 & 1
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
2 & - 2 & 0 \
0 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
2 & - 1 / 2 & 0 \
1 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & - 1 & 1 \
0 & 1 / 2 & 0 \
- 1 & 1 / 2 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 / 2 & - 1 / 2 & 0 \
0 & 1 / 2 & 0 \
- 1 / 2 & 1 / 2 & 1
\end{array} \right]$ 
E)  none 
A)  $\left[ \begin{array} { c c c } 
2 & - 2 & 0 \
0 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
2 & - 1 / 2 & 0 \
1 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & - 1 & 1 \
0 & 1 / 2 & 0 \
- 1 & 1 / 2 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 / 2 & - 1 / 2 & 0 \
0 & 1 / 2 & 0 \
- 1 / 2 & 1 / 2 & 1
\end{array} \right]$ 
E)  none D

Question 3

Solve the system of equations $\left\{ \begin{array} { r } 
x + 3 y - 2 z = 1 \
2 x + 4 y + z = 4 \
2 x + 6 y - z = 2
\end{array} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c c } 
\frac { 3 } { 3 } & \frac { 3 } { 2 } & - \frac { 11 } { 6 } \
- \frac { 2 } { 3 } & - \frac { 1 } { 2 } & \frac { 3 } { 6 } \
- \frac { 2 } { 3 } & 0 & \frac { 1 } { 3 }
\end{array} \right]$

Accepted Answer

A)  $( - 14 , - 3,0 )$ 
B)  $( 2,3,0 )$ 
C)  $( - 8,3,0 )$ 
D)  $( - 8,0,1 )$ 
E)  $( - 6 , - 3,0 )$ 
A)  $( - 14 , - 3,0 )$ 
B)  $( 2,3,0 )$ 
C)  $( - 8,3,0 )$ 
D)  $( - 8,0,1 )$ 
E)  $( - 6 , - 3,0 )$ C

Question 4

Write the system of equations for which the given matrix is the augmented matrix. $$\left[ \begin{array} { c c c c } 
1 & 2 & 6 & 3 \
- 4 & 6 & 1 & - 9 \
2 & 3 & 10 & 0
\end{array} \right]$$

Accepted Answer

\[\left\{ \begin{array} { c }

Question 5

Given $A = \left[ \begin{array} { l l } 
0 & 5
\end{array} \right]$ 
, $B = \left[ \begin{array} { l l } 
- 1 & 5 \
- 2 & 3
\end{array} \right]$ 
, $C = \left[ \begin{array} { l l l } 
0 & 1 & 2 \
6 & 2 & 3
\end{array} \right]$ 
, $D = \left[ \begin{array} { l } 
1 \
5 \
0
\end{array} \right]$ 
, find $( A B ) \cdot ( C D )$ 
, or explain why the operation cannot be performed.

Accepted Answer

None

Question 6

Find the determinant of the matrix $\left[ \begin{array} { l l } 
\frac { 3 } { 2 } & \frac { 1 } { 2 } \\
\frac { 2 } { 3 } & 2
\end{array} \right]$ , if it exists

Accepted Answer

A)  $\frac { 8 } { 5 }$ 
B)  $6$ 
C)  $- \frac { 8 } { 3 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 8 } { 3 }$ 
A)  $\frac { 8 } { 5 }$ 
B)  $6$ 
C)  $- \frac { 8 } { 3 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 8 } { 3 }$

Question 7

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } 
x + z = 3 \
y + 2 z = 1 \
- y + z = - 1
\end{array} \right.$

Accepted Answer

A)  $( 3,1,0 )$ 
B)  $( - 3 , - 1,0 )$ 
C)  $( 2,1,10 )$ 
D)  $( - 3,2,0 )$ 
E)  $( - 3,1,3 )$ 
A)  $( 3,1,0 )$ 
B)  $( - 3 , - 1,0 )$ 
C)  $( 2,1,10 )$ 
D)  $( - 3,2,0 )$ 
E)  $( - 3,1,3 )$

Question 8

Find inverse of the matrix. $\left[ \begin{array} { l l l } 
2 & 2 & 0 \
0 & 2 & 0 \
1 & 0 & 1
\end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { c c c } 
2 & - 2 & 0 \
0 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
2 & - 1 / 2 & 0 \
1 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & - 1 & 1 \
0 & 1 / 2 & 0 \
- 1 & 1 / 2 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 / 2 & - 1 / 2 & 0 \
0 & 1 / 2 & 0 \
- 1 / 2 & 1 / 2 & 1
\end{array} \right]$ 
E)  none 
A)  $\left[ \begin{array} { c c c } 
2 & - 2 & 0 \
0 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 
2 & - 1 / 2 & 0 \
1 & 2 & 0 \
- 2 & 1 / 2 & 1
\end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 
1 & - 1 & 1 \
0 & 1 / 2 & 0 \
- 1 & 1 / 2 & 1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 / 2 & - 1 / 2 & 0 \
0 & 1 / 2 & 0 \
- 1 / 2 & 1 / 2 & 1
\end{array} \right]$ 
E)  none

Question 9

Write the system of equations in matrix form. $$\left\{ \begin{array} { l } 
a - b + c = 8 \
4 a - b + c = 0 \
a + b + 2 c = 7 \
a + c + d = 0
\end{array} \right.$$

Accepted Answer

A)  $\left[ \begin{array} { c c c c } 
1 & - 1 & 1 & 0 \
4 & - 1 & 1 & 0 \
1 & 1 & 2 & 0 \
1 & 0 & 1 & 1
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c \
a
\end{array} \right] = \left[ \begin{array} { l } 
8 \
0 \
7 \
0
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c c } 
8 & - 1 & 1 & 0 \
0 & - 1 & 1 & 0 \
7 & 1 & 2 & 0 \
0 & 0 & 1 & 1
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c \
a
\end{array} \right] = \left[ \begin{array} { l } 
1 \
4 \
1 \
1
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l l l } 
8 & 1 & 1 & 0 \
0 & 4 & 1 & 0 \
7 & 1 & 2 & 0 \
0 & 1 & 1 & 1
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c \
a
\end{array} \right] = \left[ \begin{array} { l } 
0 \
0 \
0 \
1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 & - 1 & 1 \
4 & - 1 & 1 \
1 & 1 & 2
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c
\end{array} \right] = \left[ \begin{array} { l } 
8 \
0 \
7
\end{array} \right]$ 
E)  none 
A)  $\left[ \begin{array} { c c c c } 
1 & - 1 & 1 & 0 \
4 & - 1 & 1 & 0 \
1 & 1 & 2 & 0 \
1 & 0 & 1 & 1
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c \
a
\end{array} \right] = \left[ \begin{array} { l } 
8 \
0 \
7 \
0
\end{array} \right]$ 
B)  $\left[ \begin{array} { c c c c } 
8 & - 1 & 1 & 0 \
0 & - 1 & 1 & 0 \
7 & 1 & 2 & 0 \
0 & 0 & 1 & 1
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c \
a
\end{array} \right] = \left[ \begin{array} { l } 
1 \
4 \
1 \
1
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l l l } 
8 & 1 & 1 & 0 \
0 & 4 & 1 & 0 \
7 & 1 & 2 & 0 \
0 & 1 & 1 & 1
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c \
a
\end{array} \right] = \left[ \begin{array} { l } 
0 \
0 \
0 \
1
\end{array} \right]$ 
D)  $\left[ \begin{array} { c c c } 
1 & - 1 & 1 \
4 & - 1 & 1 \
1 & 1 & 2
\end{array} \right] \left[ \begin{array} { l } 
a \
b \
c
\end{array} \right] = \left[ \begin{array} { l } 
8 \
0 \
7
\end{array} \right]$ 
E)  none

Question 10

Find the determinant of the matrix $\left[ \begin{array} { l l } 
\frac { 1 } { 3 } & \frac { 1 } { 4 } \
\frac { 3 } { 2 } & 1
\end{array} \right]$ 
, if it exists.

Accepted Answer

None

Question 11

Write the system of equations for which the given matrix is the augmented matrix. $$\left[ \begin{array} { c c c c } 
0 & 2 & 6 & 3 \
1 & 0 & 0 & - 9 \
2 & 3 & 0 & 0
\end{array} \right]$$

Accepted Answer

A)  $\left\{ \begin{aligned}
y + 6 z & = 3 \
z & = - 9 \
2 x - 3 y & = 0
\end{aligned} \right.$ 
B)  $\left\{ \begin{aligned}
2 x + 6 z & = 3 \
x & = - 9 \
x + 3 y & = 0
\end{aligned} \right.$ 
C)  $\left\{ \begin{aligned}
2 x + 6 z & = 3 \
z & = - 9 \
2 y + 3 z & = 0
\end{aligned} \right.$ 
D)  $\left\{ \begin{aligned}
2 y + 6 z & = 3 \
x & = - 9 \
2 x + 3 y & = 0
\end{aligned} \right.$ 
E) none 
A)  $\left\{ \begin{aligned}
y + 6 z & = 3 \
z & = - 9 \
2 x - 3 y & = 0
\end{aligned} \right.$ 
B)  $\left\{ \begin{aligned}
2 x + 6 z & = 3 \
x & = - 9 \
x + 3 y & = 0
\end{aligned} \right.$ 
C)  $\left\{ \begin{aligned}
2 x + 6 z & = 3 \
z & = - 9 \
2 y + 3 z & = 0
\end{aligned} \right.$ 
D)  $\left\{ \begin{aligned}
2 y + 6 z & = 3 \
x & = - 9 \
2 x + 3 y & = 0
\end{aligned} \right.$ 
E) none

Question 12

Find the determinant of the matrix $\left[ \begin{array} { r r } 
\frac { 1 } { 2 } & 1 \
- \frac { 1 } { 2 } & \frac { 3 } { 2 }
\end{array} \right]$ 
, if it exists

Accepted Answer

The answer of Find the determinant of the matrix \(\left[...

Question 13

Write the system of equations in matrix form. $$\left\{ \begin{array} { l } 
8 a - 3 b + c = 7 \
4 a - 1 b + c = 1 \
a + b + 2 c = 7 \
a = 0
\end{array} \right.$$

Accepted Answer

\[\left[ \begin{array} { c c c c } 
8 &

Question 14

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$, $( - 1 , - 6 )$, and $( 3 , - 6 )$

Accepted Answer

A)  $y = x ^ { 2 } + x$ 
B)  $y = x ^ { 2 } - 4 x + 2$ 
C)  $y = x ^ { 2 } + 2 x - 1$ 
D)  $y = 2 x ^ { 2 } + 4$ 
E)  $y = - 2 x ^ { 2 } + 4 x$ 
A)  $y = x ^ { 2 } + x$ 
B)  $y = x ^ { 2 } - 4 x + 2$ 
C)  $y = x ^ { 2 } + 2 x - 1$ 
D)  $y = 2 x ^ { 2 } + 4$ 
E)  $y = - 2 x ^ { 2 } + 4 x$

Question 15

Given $A = \left[ \begin{array} { c c } 
3 & - 1 \
2 & 2
\end{array} \right]$ 
and $B = \left[ \begin{array} { c c } 
2 & 3 \
- 1 & 5
\end{array} \right]$ 
, find $A B$ , or explain why the operation cannot be performed.

Accepted Answer

None

Question 16

Use Cramer's Rule to solve the system. $$\left\{ \begin{array} { r } 
2 x - y - 3 z = 3 \
x + 3 y - 2 z = - 4 \
7 x + 7 y - 12 z = - 5
\end{array} \right.$$

Accepted Answer

A)  $( 3,1,0 )$ 
B)  $( - 3 , - 1,0 )$ 
C)  $( 2,1,10 )$ 
D)  $( - 3,2,0 )$ 
E) no solution 
A)  $( 3,1,0 )$ 
B)  $( - 3 , - 1,0 )$ 
C)  $( 2,1,10 )$ 
D)  $( - 3,2,0 )$ 
E) no solution

Question 17

Find the complete solution of the system. $$\left\{ \begin{array} { r } 
x + z = 3 \
y + 2 z = 1 \
- y + z = - 1
\end{array} \right.$$

Accepted Answer

The answer of Find the complete solution of the system....

Question 18

Write out the form of the partial fraction decomposition of $$\frac { x ^ { 2 } - 8 } { ( x - 1 ) ^ { 2 } ( x + 3 ) }$$ 
.

Accepted Answer

The answer of Write out the form of the partial...

Question 19

Find the dimension of matrix A. $$A = \left[ \begin{array} { c c c } 
4 & 1 & - 1 \
3 & - 5 & 2
\end{array} \right]$$

Accepted Answer

The answer of Find the dimension of matrix A. \[A...

Question 20

Solve $\left[ \begin{array} { l l } 
2 & x \
6 & 0
\end{array} \right] \left[ \begin{array} { l l } 
y & 3 \
- 2 & 2
\end{array} \right] = \left[ \begin{array} { l l } 
1 & 3 \
- 6 & 18
\end{array} \right]$ 
for $x$ 
and $y$ 
.

Accepted Answer

None

Find the complete solution of the system. $\left\{ \begin{array} { r } x + y + z = 3 \\x - y + z = - 2 \\x + y + 3 z = 1\end{array} \right.$

Find inverse of the matrix. $\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]$

Write the system of equations for which the given matrix is the augmented matrix. $\left[ \begin{array} { c c c c } 1 & 2 & 6 & 3 \\- 4 & 6 & 1 & - 9 \\2 & 3 & 10 & 0\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { l l } \frac { 3 } { 2 } & \frac { 1 } { 2 } \\\\\frac { 2 } { 3 } & 2\end{array} \right]$ , if it exists

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + z = 3 \\y + 2 z = 1 \\- y + z = - 1\end{array} \right.$

Find inverse of the matrix. $\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]$

Write the system of equations in matrix form. $\left\{ \begin{array} { l } a - b + c = 8 \\4 a - b + c = 0 \\a + b + 2 c = 7 \\a + c + d = 0\end{array} \right.$

Find the determinant of the matrix $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 1 } { 4 } \\\frac { 3 } { 2 } & 1\end{array} \right]$ , if it exists.

Write the system of equations for which the given matrix is the augmented matrix. $\left[ \begin{array} { c c c c } 0 & 2 & 6 & 3 \\1 & 0 & 0 & - 9 \\2 & 3 & 0 & 0\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { r r } \frac { 1 } { 2 } & 1 \\- \frac { 1 } { 2 } & \frac { 3 } { 2 }\end{array} \right]$ , if it exists

Write the system of equations in matrix form. $\left\{ \begin{array} { l } 8 a - 3 b + c = 7 \\4 a - 1 b + c = 1 \\a + b + 2 c = 7 \\a = 0\end{array} \right.$

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$ , $( - 1 , - 6 )$ , and $( 3 , - 6 )$

Given $A = \left[ \begin{array} { c c } 3 & - 1 \\2 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c } 2 & 3 \\- 1 & 5\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } 2 x - y - 3 z = 3 \\x + 3 y - 2 z = - 4 \\7 x + 7 y - 12 z = - 5\end{array} \right.$

Find the complete solution of the system. $\left\{ \begin{array} { r } x + z = 3 \\y + 2 z = 1 \\- y + z = - 1\end{array} \right.$

Write out the form of the partial fraction decomposition of $\frac { x ^ { 2 } - 8 } { ( x - 1 ) ^ { 2 } ( x + 3 ) }$ .

Find the dimension of matrix A. $A = \left[ \begin{array} { c c c } 4 & 1 & - 1 \\3 & - 5 & 2\end{array} \right]$

Solve $\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right]$ for $x$ and $y$ .

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Counting and Probability

Filters

Exam 11: Matrices and Determinants

Find the complete solution of the system. {x+y+z=3x−y+z=−2x+y+3z=1\left\{ \begin{array} { r } x + y + z = 3 \\x - y + z = - 2 \\x + y + 3 z = 1\end{array} \right.⎩⎨⎧​x+y+z=3x−y+z=−2x+y+3z=1​

Find inverse of the matrix. [220020101]\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]​201​220​001​​

Write the system of equations for which the given matrix is the augmented matrix. [1263−461−923100]\left[ \begin{array} { c c c c } 1 & 2 & 6 & 3 \\- 4 & 6 & 1 & - 9 \\2 & 3 & 10 & 0\end{array} \right]​1−42​263​6110​3−90​​

Find the determinant of the matrix [3212232]\left[ \begin{array} { l l } \frac { 3 } { 2 } & \frac { 1 } { 2 } \\\\\frac { 2 } { 3 } & 2\end{array} \right]​23​32​​21​2​​ , if it exists

Use Cramer's Rule to solve the system. {x+z=3y+2z=1−y+z=−1\left\{ \begin{array} { r } x + z = 3 \\y + 2 z = 1 \\- y + z = - 1\end{array} \right.⎩⎨⎧​x+z=3y+2z=1−y+z=−1​

Find inverse of the matrix. [220020101]\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]​201​220​001​​

Write the system of equations in matrix form. {a−b+c=84a−b+c=0a+b+2c=7a+c+d=0\left\{ \begin{array} { l } a - b + c = 8 \\4 a - b + c = 0 \\a + b + 2 c = 7 \\a + c + d = 0\end{array} \right.⎩⎨⎧​a−b+c=84a−b+c=0a+b+2c=7a+c+d=0​

Find the determinant of the matrix [1314321]\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 1 } { 4 } \\\frac { 3 } { 2 } & 1\end{array} \right][31​23​​41​1​] , if it exists.

Write the system of equations for which the given matrix is the augmented matrix. [0263100−92300]\left[ \begin{array} { c c c c } 0 & 2 & 6 & 3 \\1 & 0 & 0 & - 9 \\2 & 3 & 0 & 0\end{array} \right]​012​203​600​3−90​​

Find the determinant of the matrix [121−1232]\left[ \begin{array} { r r } \frac { 1 } { 2 } & 1 \\- \frac { 1 } { 2 } & \frac { 3 } { 2 }\end{array} \right][21​−21​​123​​] , if it exists

Write the system of equations in matrix form. {8a−3b+c=74a−1b+c=1a+b+2c=7a=0\left\{ \begin{array} { l } 8 a - 3 b + c = 7 \\4 a - 1 b + c = 1 \\a + b + 2 c = 7 \\a = 0\end{array} \right.⎩⎨⎧​8a−3b+c=74a−1b+c=1a+b+2c=7a=0​

Find the quadratic polynomial y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c whose graph passes through the points (−2,−16)( - 2 , - 16 )(−2,−16) , (−1,−6)( - 1 , - 6 )(−1,−6) , and (3,−6)( 3 , - 6 )(3,−6)

Given A=[3−122]A = \left[ \begin{array} { c c } 3 & - 1 \\2 & 2\end{array} \right]A=[32​−12​] and B=[23−15]B = \left[ \begin{array} { c c } 2 & 3 \\- 1 & 5\end{array} \right]B=[2−1​35​] , find ABA BAB , or explain why the operation cannot be performed.

Use Cramer's Rule to solve the system. {2x−y−3z=3x+3y−2z=−47x+7y−12z=−5\left\{ \begin{array} { r } 2 x - y - 3 z = 3 \\x + 3 y - 2 z = - 4 \\7 x + 7 y - 12 z = - 5\end{array} \right.⎩⎨⎧​2x−y−3z=3x+3y−2z=−47x+7y−12z=−5​

Find the complete solution of the system. {x+z=3y+2z=1−y+z=−1\left\{ \begin{array} { r } x + z = 3 \\y + 2 z = 1 \\- y + z = - 1\end{array} \right.⎩⎨⎧​x+z=3y+2z=1−y+z=−1​

Write out the form of the partial fraction decomposition of x2−8(x−1)2(x+3)\frac { x ^ { 2 } - 8 } { ( x - 1 ) ^ { 2 } ( x + 3 ) }(x−1)2(x+3)x2−8​ .

Find the dimension of matrix A. A=[41−13−52]A = \left[ \begin{array} { c c c } 4 & 1 & - 1 \\3 & - 5 & 2\end{array} \right]A=[43​1−5​−12​]

Solve [2x60][y3−22]=[13−618]\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right][26​x0​][y−2​32​]=[1−6​318​] for xxx and yyy .

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Counting and Probability

Filters

Find the complete solution of the system. $\left\{ \begin{array} { r } x + y + z = 3 \\x - y + z = - 2 \\x + y + 3 z = 1\end{array} \right.$

Find inverse of the matrix. $\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]$

Write the system of equations for which the given matrix is the augmented matrix. $\left[ \begin{array} { c c c c } 1 & 2 & 6 & 3 \\- 4 & 6 & 1 & - 9 \\2 & 3 & 10 & 0\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { l l } \frac { 3 } { 2 } & \frac { 1 } { 2 } \\\\\frac { 2 } { 3 } & 2\end{array} \right]$ , if it exists

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + z = 3 \\y + 2 z = 1 \\- y + z = - 1\end{array} \right.$

Find inverse of the matrix. $\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]$

Write the system of equations in matrix form. $\left\{ \begin{array} { l } a - b + c = 8 \\4 a - b + c = 0 \\a + b + 2 c = 7 \\a + c + d = 0\end{array} \right.$

Find the determinant of the matrix $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 1 } { 4 } \\\frac { 3 } { 2 } & 1\end{array} \right]$ , if it exists.

Write the system of equations for which the given matrix is the augmented matrix. $\left[ \begin{array} { c c c c } 0 & 2 & 6 & 3 \\1 & 0 & 0 & - 9 \\2 & 3 & 0 & 0\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { r r } \frac { 1 } { 2 } & 1 \\- \frac { 1 } { 2 } & \frac { 3 } { 2 }\end{array} \right]$ , if it exists

Write the system of equations in matrix form. $\left\{ \begin{array} { l } 8 a - 3 b + c = 7 \\4 a - 1 b + c = 1 \\a + b + 2 c = 7 \\a = 0\end{array} \right.$

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$ , $( - 1 , - 6 )$ , and $( 3 , - 6 )$

Given $A = \left[ \begin{array} { c c } 3 & - 1 \\2 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c } 2 & 3 \\- 1 & 5\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } 2 x - y - 3 z = 3 \\x + 3 y - 2 z = - 4 \\7 x + 7 y - 12 z = - 5\end{array} \right.$

Find the complete solution of the system. $\left\{ \begin{array} { r } x + z = 3 \\y + 2 z = 1 \\- y + z = - 1\end{array} \right.$

Write out the form of the partial fraction decomposition of $\frac { x ^ { 2 } - 8 } { ( x - 1 ) ^ { 2 } ( x + 3 ) }$ .

Find the dimension of matrix A. $A = \left[ \begin{array} { c c c } 4 & 1 & - 1 \\3 & - 5 & 2\end{array} \right]$

Solve $\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right]$ for $x$ and $y$ .