Question 1

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

$\left[ \begin{array} { l l } 
2 & x \
6 & 0
\end{array} \right] \left[ \begin{array} { l l } 
y & 3 \
- 2 & 2
\end{array} \right] = \left[ \begin{array} { c c } 
2 y - 2 x & 6 + 2 x \
6 y & 18
\end{array} \right]$   $\Rightarrow$

$\left[ \begin{array} { c c } 
2 y - 2 x & 6 + 2 x \
6 y & 18
\end{array} \right] = \left[ \begin{array} { c c } 
1 & 3 \
- 6 & 18
\end{array} \right]$ $\Rightarrow$

$\left\{ \begin{array} { l } 
2 y - 2 x = 1 \
6 + 2 x = 3
\end{array} \right.$   $\Rightarrow$   $x = - \frac { 3 } { 2 }$  , $y = - 1$

Question 2

Find the complete solution of the system. $$\left\{ \begin{array} { r } 
2 x + 3 y = 11 \
3 x + 3 y = 9 \
x + z = 1
\end{array} \right.$$

Accepted Answer

$x = - 2$ 
, $y = 5$ 
, $z = 3$

Question 3

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

$x = 3$ 
, $y = 1$ 
, $z = 0$

Question 4

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

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

Question 5

Find the determinant and, if possible, the inverse of the matrix. $\left[ \begin{array} { c c c } 
1 & 0 & 0 \
1 & 3 & 1 \
- 1 & - 2 & 0
\end{array} \right]$

Accepted Answer

The answer of Find the determinant and, if possible, the...

Question 6

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } 
x + 2 y - 4 z = 0 \
2 x + 5 y = 4 \
y - z = 3
\end{array} \right.$

Accepted Answer

The answer of Use Cramer's Rule to solve the system....

Question 7

A man invests a total of $\$ 21,000$  in two accounts, one paying $5.5 \%$  simple interest per year and the other paying $8.5 \%$  simple interest per year. His annual interest from both accounts is $\$ 1,280$  . How much did he invest at each rate?

Accepted Answer

The answer of A man invests a total of \(\$...

Question 8

Find the complete solution of the system. $$\left\{ \begin{array} { r } 
5 x + 3 y + 11 z = 37 \
3 y + z = 12 \
15 x + 9 y + 13 z = 61
\end{array} \right.$$

Accepted Answer

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

Question 9

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 10

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

Accepted Answer

The answer of Write the system of equations for which...

Question 11

Solve the system. $\left\{ \begin{array} { l } 
x = y ^ { 2 } \
x - y = 2
\end{array} \right.$

Accepted Answer

The answer of Solve the system. \(\left\{ \begin{array} { l...

Question 12

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

The answer of Given \(A = \left[ \begin{array} { c...

Question 13

Solve the system of equations $\left\{ \begin{array} { r } 
3 x + 2 y = 8 \
x - 2 y = 0
\end{array} \right.$ 
by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c } 
\frac { 1 } { 4 } & \frac { 1 } { 4 } \
\frac { 1 } { 3 } & - \frac { 3 } { 3 }
\end{array} \right]$  .

Accepted Answer

The answer of Solve the system of equations \(\left\{ \begin{array}...

Question 14

Determine whether the system is linear or nonlinear. $$\left\{ \begin{array} { l } 
y = 3 - 2 x \
y - 4 = 2 x
\end{array} \right.$$

Accepted Answer

The answer of Determine whether the system is linear or...

Question 15

Solve the system. $\left\{ \begin{array} { l } 
x + y = 6 \
x - 2 y = 0
\end{array} \right.$

Accepted Answer

The answer of Solve the system. \(\left\{ \begin{array} { l...

Question 16

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

The answer of Write the system of equations in matrix...

Question 17

Given $A = \left[ \begin{array} { c c } 
- 9 & 8 \
- 6 & - 5 \
10 & - 16
\end{array} \right]$ and $B = \left[ \begin{array} { l l } 
1 & 0 \
2 & 3 \
0 & 4
\end{array} \right]$

, solve $2 X - A = 3 B$ , for $X$  , or explain why there is no solution.

Accepted Answer

The answer of Given \(A = \left[ \begin{array} { c...

Question 18

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

The answer of Given \(A = \left[ \begin{array} { l...

Question 19

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

Accepted Answer

The answer of Find the quadratic polynomial \(y = ax...

Question 20

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither. $$\left[ \begin{array} { l l l l } 
1 & 1 & 0 & 3 \
0 & 1 & 0 & 3 \
2 & 0 & 1 & 1
\end{array} \right]$$

Accepted Answer

The answer of Determine whether the matrix is in row-echelon...

Quiz 11: Matrices and Determinants

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 $y$ .

Find the complete solution of the system. $\left\{ \begin{array} { r } 2 x + 3 y = 11 \\3 x + 3 y = 9 \\x + z = 1\end{array} \right.$

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

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.

Find the determinant and, if possible, the inverse of the matrix. $\left[ \begin{array} { c c c } 1 & 0 & 0 \\1 & 3 & 1 \\- 1 & - 2 & 0\end{array} \right]$

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.$

A man invests a total of $\$ 21,000$ in two accounts, one paying $5.5 \%$ simple interest per year and the other paying $8.5 \%$ simple interest per year. His annual interest from both accounts is $\$ 1,280$ . How much did he invest at each rate?

Find the complete solution of the system. $\left\{ \begin{array} { r } 5 x + 3 y + 11 z = 37 \\3 y + z = 12 \\15 x + 9 y + 13 z = 61\end{array} \right.$

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

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

Solve the system. $\left\{ \begin{array} { l } x = y ^ { 2 } \\x - y = 2\end{array} \right.$

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.

Determine whether the system is linear or nonlinear. $\left\{ \begin{array} { l } y = 3 - 2 x \\y - 4 = 2 x\end{array} \right.$

Solve the system. $\left\{ \begin{array} { l } x + y = 6 \\x - 2 y = 0\end{array} \right.$

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

Given $A = \left[ \begin{array} { c c } - 9 & 8 \\- 6 & - 5 \\10 & - 16\end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \\2 & 3 \\0 & 4\end{array} \right]$ , solve $2 X - A = 3 B$ , for $X$ , or explain why there is no solution.

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

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither. $\left[ \begin{array} { l l l l } 1 & 1 & 0 & 3 \\0 & 1 & 0 & 3 \\2 & 0 & 1 & 1\end{array} \right]$

Quiz 11: Matrices and Determinants

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 .

Find the complete solution of the system. {2x+3y=113x+3y=9x+z=1\left\{ \begin{array} { r } 2 x + 3 y = 11 \\3 x + 3 y = 9 \\x + z = 1\end{array} \right.⎩⎨⎧​2x+3y=113x+3y=9x+z=1​

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​

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.

Find the determinant and, if possible, the inverse of the matrix. [100131−1−20]\left[ \begin{array} { c c c } 1 & 0 & 0 \\1 & 3 & 1 \\- 1 & - 2 & 0\end{array} \right]​11−1​03−2​010​​

Use Cramer's Rule to solve the system. {x+2y−4z=02x+5y=4y−z=3\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.⎩⎨⎧​x+2y−4z=02x+5y=4y−z=3​

A man invests a total of $21,000\$ 21,000$21,000 in two accounts, one paying 5.5%5.5 \%5.5% simple interest per year and the other paying 8.5%8.5 \%8.5% simple interest per year. His annual interest from both accounts is $1,280\$ 1,280$1,280 . How much did he invest at each rate?

Find the complete solution of the system. {5x+3y+11z=373y+z=1215x+9y+13z=61\left\{ \begin{array} { r } 5 x + 3 y + 11 z = 37 \\3 y + z = 12 \\15 x + 9 y + 13 z = 61\end{array} \right.⎩⎨⎧​5x+3y+11z=373y+z=1215x+9y+13z=61​

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​]

Write the system of equations for which the given matrix is the augmented matrix. [136306092−300]\left[ \begin{array} { c c c c } 1 & 3 & 6 & 3 \\0 & 6 & 0 & 9 \\2 & - 3 & 0 & 0\end{array} \right]​102​36−3​600​390​​

Solve the system. {x=y2x−y=2\left\{ \begin{array} { l } x = y ^ { 2 } \\x - y = 2\end{array} \right.{x=y2x−y=2​

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.

Determine whether the system is linear or nonlinear. {y=3−2xy−4=2x\left\{ \begin{array} { l } y = 3 - 2 x \\y - 4 = 2 x\end{array} \right.{y=3−2xy−4=2x​

Solve the system. {x+y=6x−2y=0\left\{ \begin{array} { l } x + y = 6 \\x - 2 y = 0\end{array} \right.{x+y=6x−2y=0​

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 = ax ^ { 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) .

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither. [110301032011]\left[ \begin{array} { l l l l } 1 & 1 & 0 & 3 \\0 & 1 & 0 & 3 \\2 & 0 & 1 & 1\end{array} \right]​102​110​001​331​​

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 $y$ .

Find the complete solution of the system. $\left\{ \begin{array} { r } 2 x + 3 y = 11 \\3 x + 3 y = 9 \\x + z = 1\end{array} \right.$

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

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.

Find the determinant and, if possible, the inverse of the matrix. $\left[ \begin{array} { c c c } 1 & 0 & 0 \\1 & 3 & 1 \\- 1 & - 2 & 0\end{array} \right]$

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.$

A man invests a total of $\$ 21,000$ in two accounts, one paying $5.5 \%$ simple interest per year and the other paying $8.5 \%$ simple interest per year. His annual interest from both accounts is $\$ 1,280$ . How much did he invest at each rate?

Find the complete solution of the system. $\left\{ \begin{array} { r } 5 x + 3 y + 11 z = 37 \\3 y + z = 12 \\15 x + 9 y + 13 z = 61\end{array} \right.$

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

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

Solve the system. $\left\{ \begin{array} { l } x = y ^ { 2 } \\x - y = 2\end{array} \right.$

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.

Determine whether the system is linear or nonlinear. $\left\{ \begin{array} { l } y = 3 - 2 x \\y - 4 = 2 x\end{array} \right.$

Solve the system. $\left\{ \begin{array} { l } x + y = 6 \\x - 2 y = 0\end{array} \right.$

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

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither. $\left[ \begin{array} { l l l l } 1 & 1 & 0 & 3 \\0 & 1 & 0 & 3 \\2 & 0 & 1 & 1\end{array} \right]$