Question 1

Solve using Cramerʹs rule.
-$$\begin{array} { l } 
3 x _ { 1 } + 3 x _ { 2 } = 9 \
2 x _ { 1 } + x _ { 2 } = 1
\end{array}$$

Accepted Answer

A)  $( 2 , - 5 )$ 
B)  $( 5 , - 2 )$ 
C)  $( - 5 , - 2 )$ 
D)  $( - 2,5 )$ 
A)  $( 2 , - 5 )$ 
B)  $( 5 , - 2 )$ 
C)  $( - 5 , - 2 )$ 
D)  $( - 2,5 )$ D

Question 2

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r } 
1 & 0 & 0 \
6 & - 5 & 0 \
- 5 & 2 & 3
\end{array} \right]$$

Accepted Answer

A)  3 
B)  $- 27$ 
C)  15 
D)  $- 15$ 
A)  3 
B)  $- 27$ 
C)  15 
D)  $- 15$ D

Question 3

Determine the values of the parameter s for which the system has a unique solution, and describe the solution.
-$$\begin{array} { l } 
s _ { 1 } - 5 s _ { 2 } = 3 \
2 x _ { 1 } - 10 s _ { 2 } = 5
\end{array}$$

Accepted Answer

A)  $s \neq 0,1 ; x _ { 1 } = \frac { 1 } { 2 ( s - 1 ) }$ and $x _ { 2 } = \frac { 6 - 5 s } { 10 s ( s - 1 ) }$ 
B)  $s \neq 1 ; x _ { 1 } = \frac { 11 } { 2 ( s + 1 ) }$ and $x _ { 2 } = \frac { 6 + 5 s } { 10 s ( s + 1 ) }$ 
C)  $s \neq 1 ; x _ { 1 } = \frac { 1 } { 10 ( s - 1 ) ( s + 1 ) }$ and $x _ { 2 } = \frac { 6 - 5 s } { 10 ( s - 1 ) ( s + 1 ) }$ 
D)  $s \neq \pm 1 ; x _ { 1 } = \frac { 1 } { 2 ( s + 1 ) }$ and $x _ { 2 } = \frac { 6 - 5 s } { 10 s ( s + 1 ) }$ 
A)  $s \neq 0,1 ; x _ { 1 } = \frac { 1 } { 2 ( s - 1 ) }$ and $x _ { 2 } = \frac { 6 - 5 s } { 10 s ( s - 1 ) }$ 
B)  $s \neq 1 ; x _ { 1 } = \frac { 11 } { 2 ( s + 1 ) }$ and $x _ { 2 } = \frac { 6 + 5 s } { 10 s ( s + 1 ) }$ 
C)  $s \neq 1 ; x _ { 1 } = \frac { 1 } { 10 ( s - 1 ) ( s + 1 ) }$ and $x _ { 2 } = \frac { 6 - 5 s } { 10 ( s - 1 ) ( s + 1 ) }$ 
D)  $s \neq \pm 1 ; x _ { 1 } = \frac { 1 } { 2 ( s + 1 ) }$ and $x _ { 2 } = \frac { 6 - 5 s } { 10 s ( s + 1 ) }$ A

Question 4

Calculate the area of the parallelogram with the given vertices.
-$$( 0,0 ) , ( 5,8 ) , ( 11,12 ) , ( 6,4 )$$

Accepted Answer

A)  38 
B)  56 
C)  27 
D)  28 
A)  38 
B)  56 
C)  27 
D)  28

Question 5

Solve using Cramerʹs rule.
-$$\begin{array} { l } 
5 x _ { 1 } + 9 x _ { 2 } = - 3 \
5 x _ { 1 } + s _ { 2 } = 4
\end{array}$$

Accepted Answer

A)  $s \neq \pm 5 ; x _ { 1 } = \frac { - 3 s - 36 } { 5 ( s - 5 ) ( s + 5 ) }$ and $x _ { 2 } = \frac { 4 s + 3 } { 5 ( s - 5 ) ( s + 5 ) }$ 
B)  $s \neq 3 ; x _ { 1 } = \frac { - 3 s + 36 } { 5 ( s - 3 ) ( s + 3 ) }$ and $x _ { 2 } = \frac { 4 s - 3 } { ( s - 3 ) ( s + 3 ) }$ 
C)  $s \neq \pm 3 ; x _ { 1 } = \frac { - 3 s - 36 } { 5 ( s - 3 ) ( s + 3 ) }$ and $x _ { 2 } = \frac { 4 s + 3 } { ( s - 3 ) ( s + 3 ) }$ 
D)  $s \neq \pm 9 ; x _ { 1 } = - 3 s - 36$ and $x _ { 2 } = 4 s + 3$ 
A)  $s \neq \pm 5 ; x _ { 1 } = \frac { - 3 s - 36 } { 5 ( s - 5 ) ( s + 5 ) }$ and $x _ { 2 } = \frac { 4 s + 3 } { 5 ( s - 5 ) ( s + 5 ) }$ 
B)  $s \neq 3 ; x _ { 1 } = \frac { - 3 s + 36 } { 5 ( s - 3 ) ( s + 3 ) }$ and $x _ { 2 } = \frac { 4 s - 3 } { ( s - 3 ) ( s + 3 ) }$ 
C)  $s \neq \pm 3 ; x _ { 1 } = \frac { - 3 s - 36 } { 5 ( s - 3 ) ( s + 3 ) }$ and $x _ { 2 } = \frac { 4 s + 3 } { ( s - 3 ) ( s + 3 ) }$ 
D)  $s \neq \pm 9 ; x _ { 1 } = - 3 s - 36$ and $x _ { 2 } = 4 s + 3$

Question 6

Compute the determinant of the matrix by cofactor expansion.
-$\left[ \begin{array} { l l l } 3 & 2 & 5 \ 1 & 1 & 4 \ 3 & 3 & 4 \end{array} \right]$

Accepted Answer

A)  $- 56$ 
B)  110 
C)  8 
D)  $- 8$ 
A)  $- 56$ 
B)  110 
C)  8 
D)  $- 8$

Question 7

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r } 
3 & 1 & 2 \
- 3 & - 1 & - 4 \
6 & 5 & 3
\end{array} \right]$$

Accepted Answer

A)  0 
B)  $- 9$ 
C)  18 
D)  $- 18$ 
A)  0 
B)  $- 9$ 
C)  18 
D)  $- 18$

Question 8

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { l l l } 
4 & 2 & 9 \
5 & 2 & 9 \
7 & 7 & 4
\end{array} \right]$$

Accepted Answer

A)  55 
B)  891 
C)  $- 197$ 
D)  $- 55$ 
A)  55 
B)  891 
C)  $- 197$ 
D)  $- 55$

Question 9

Compute the determinant of the matrix by cofactor expansion.
-$\left[ \begin{array} { r r r } 5 & 1 & 3 \ 1 & - 2 & 2 \ 3 & 5 & 4 \end{array} \right]$

Accepted Answer

A)  $- 55$ 
B)  $- 67$ 
C)  17 
D)  55 
A)  $- 55$ 
B)  $- 67$ 
C)  17 
D)  55

Question 10

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r r r } 
2 & 2 & - 2 & 5 & - 4 \
0 & 3 & 2 & 1 & 0 \
0 & 0 & - 2 & 3 & 7 \
0 & 0 & 0 & - 1 & - 5 \
0 & 0 & 0 & 0 & 2
\end{array} \right]$$

Accepted Answer

A)  24 
B)  $- 24$ 
C)  $- 12$ 
D)  0 
A)  24 
B)  $- 24$ 
C)  $- 12$ 
D)  0

Question 11

Compute the determinant of the matrix by cofactor expansion.
-$\left[ \begin{array} { r r r } 4 & - 8 & 2 \ 0 & 1 & 8 \ 0 & 0 & - 9 \end{array} \right]$

Accepted Answer

A)  $- 100$ 
B)  $- 28$ 
C)  $- 36$ 
D)  36 
A)  $- 100$ 
B)  $- 28$ 
C)  $- 36$ 
D)  36

Question 12

Calculate the area of the parallelogram with the given vertices.
-$$( - 1 , - 2 ) , ( 1,4 ) , ( 6,2 ) , ( 8,8 )$$

Accepted Answer

A)  68 
B)  34 
C)  33 
D)  38 
A)  68 
B)  34 
C)  33 
D)  38

Question 13

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r r } 
2 & 1 & - 3 & 1 \
0 & 5 & - 3 & 5 \
- 4 & 3 & 3 & 3 \
- 2 & 5 & 1 & 3
\end{array} \right]$$

Accepted Answer

A)  1 
B)  0 
C)  27 
D)  $- 1$ 
A)  1 
B)  0 
C)  27 
D)  $- 1$

Question 14

Determine the values of the parameter s for which the system has a unique solution, and describe the solution.
-$$\begin{array} { l } 
2 x _ { 1 } + 2 x _ { 2 } = 26 \
2 x _ { 1 } - 2 x _ { 2 } = - 10
\end{array}$$

Accepted Answer

A)  $( 4,9 )$ 
B)  $( - 4 , - 9 )$ 
C)  $( 9,4 )$ 
D)  $( - 9,4 )$ 
A)  $( 4,9 )$ 
B)  $( - 4 , - 9 )$ 
C)  $( 9,4 )$ 
D)  $( - 9,4 )$

Question 15

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r r } 
2 & - 2 & 6 & 4 \
2 & 2 & 7 & 1 \
6 & - 6 & 13 & 14 \
- 2 & 2 & - 6 & 2
\end{array} \right]$$

Accepted Answer

A)  -60 
B)  $- 240$ 
C)  120 
D)  $- 120$ 
A)  -60 
B)  $- 240$ 
C)  120 
D)  $- 120$

Question 16

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r } 
1 & 2 & 5 \
4 & 8 & 12 \
- 2 & 3 & - 1
\end{array} \right]$$

Accepted Answer

A)  14 
B)  56 
C)  $- 28$ 
D)  28 
A)  14 
B)  56 
C)  $- 28$ 
D)  28

Question 17

Compute the determinant of the matrix by cofactor expansion.
-$$\left[ \begin{array} { r r r r } 
- 3 & 2 & - 2 & 2 \
0 & - 1 & 2 & - 2 \
0 & 3 & 0 & 0 \
0 & - 3 & 1 & 4
\end{array} \right]$$

Accepted Answer

A)  0 
B)  $- 90$ 
C)  $- 30$ 
D)  90 
A)  0 
B)  $- 90$ 
C)  $- 30$ 
D)  90

Question 18

Solve using Cramerʹs rule.
-$$\begin{array} { l } 
2 x _ { 1 } + 3 x _ { 2 } = 26 \
- 2 x _ { 1 } + 4 x _ { 2 } = 2
\end{array}$$

Accepted Answer

A)  $( - 4,7 )$ 
B)  $( 7,4 )$ 
C)  $( 4,7 )$ 
D)  $( - 7 , - 4 )$ 
A)  $( - 4,7 )$ 
B)  $( 7,4 )$ 
C)  $( 4,7 )$ 
D)  $( - 7 , - 4 )$

Solve using Cramerʹs rule. - 3+3=9 2+=1

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r } 1 & 0 & 0 \\6 & - 5 & 0 \\- 5 & 2 & 3\end{array} \right]$

Determine the values of the parameter s for which the system has a unique solution, and describe the solution. - -5=3 2-10=5

Calculate the area of the parallelogram with the given vertices. - $( 0,0 ) , ( 5,8 ) , ( 11,12 ) , ( 6,4 )$

Solve using Cramerʹs rule. - 5+9=-3 5+=4

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { l l l } 3 & 2 & 5 \\ 1 & 1 & 4 \\ 3 & 3 & 4 \end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r } 3 & 1 & 2 \\- 3 & - 1 & - 4 \\6 & 5 & 3\end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { l l l } 4 & 2 & 9 \\5 & 2 & 9 \\7 & 7 & 4\end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r } 5 & 1 & 3 \\ 1 & - 2 & 2 \\ 3 & 5 & 4 \end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r r r } 2 & 2 & - 2 & 5 & - 4 \\0 & 3 & 2 & 1 & 0 \\0 & 0 & - 2 & 3 & 7 \\0 & 0 & 0 & - 1 & - 5 \\0 & 0 & 0 & 0 & 2\end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r } 4 & - 8 & 2 \\ 0 & 1 & 8 \\ 0 & 0 & - 9 \end{array} \right]$

Calculate the area of the parallelogram with the given vertices. - $( - 1 , - 2 ) , ( 1,4 ) , ( 6,2 ) , ( 8,8 )$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r r } 2 & 1 & - 3 & 1 \\0 & 5 & - 3 & 5 \\- 4 & 3 & 3 & 3 \\- 2 & 5 & 1 & 3\end{array} \right]$

Determine the values of the parameter s for which the system has a unique solution, and describe the solution. - 2+2=26 2-2=-10

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r r } 2 & - 2 & 6 & 4 \\2 & 2 & 7 & 1 \\6 & - 6 & 13 & 14 \\- 2 & 2 & - 6 & 2\end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r } 1 & 2 & 5 \\4 & 8 & 12 \\- 2 & 3 & - 1\end{array} \right]$

Compute the determinant of the matrix by cofactor expansion. - $\left[ \begin{array} { r r r r } - 3 & 2 & - 2 & 2 \\0 & - 1 & 2 & - 2 \\0 & 3 & 0 & 0 \\0 & - 3 & 1 & 4\end{array} \right]$

Solve using Cramerʹs rule. - 2+3=26 -2+4=2

Linear Equations in Linear Algebra

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Exam 3: Determinants

Solve using Cramerʹs rule. - 3+3=9 2+=1

Compute the determinant of the matrix by cofactor expansion. - [1006−50−523]\left[ \begin{array} { r r r } 1 & 0 & 0 \\6 & - 5 & 0 \\- 5 & 2 & 3\end{array} \right]​16−5​0−52​003​​

Determine the values of the parameter s for which the system has a unique solution, and describe the solution. - -5=3 2-10=5

Calculate the area of the parallelogram with the given vertices. - (0,0),(5,8),(11,12),(6,4)( 0,0 ) , ( 5,8 ) , ( 11,12 ) , ( 6,4 )(0,0),(5,8),(11,12),(6,4)