Question 1

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 
- 4 & 0 \
5 & 8
\end{array} \right]$$ ​

Accepted Answer

A) 32 
B)  $- 32$ 
C)  $- \frac { 1 } { 32 }$ 
D) 0 
E)  $\frac { 1 } { 32 }$ 
A) 32 
B)  $- 32$ 
C)  $- \frac { 1 } { 32 }$ 
D) 0 
E)  $\frac { 1 } { 32 }$

Question 2

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $$\left[ \begin{array} { c c c c c } 
- 1 & 1 & 0 & - 2 & 3 \
0 & 0 & 1 & 4 & 0 \
0 & - 2 & - 3 & - 5 & 1 \
1 & 0 & 0 & 7 & 0 \
0 & 0 & 0 & 1 & 0
\end{array} \right]$$ .

Accepted Answer

A) -12 
B) -10 
C) 7 
D) -6 
E) 4 
A) -12 
B) -10 
C) 7 
D) -6 
E) 4

Question 3

Evaluate the determinant.​ $$\left| \begin{array} { c c c c } 
5 & 5 & - 4 & 2 \
4 & 1 & - 4 & 1 \
- 1 & 2 & 1 & - 4 \
1 & 2 & - 5 & - 1
\end{array} \right|$$

Accepted Answer

The answer of Evaluate the determinant.​ \[\left| \begin{array} { c...

Question 4

Evaluate the determinant $$\left| \begin{array} { c c } 
4 x & x \ln x \
4 & 6 + \ln x
\end{array} \right|$$ in which the entries are functions.

Accepted Answer

A)  $16 \ln x$ 
B)  $4 - \ln x$ 
C)  $6 \ln x$ 
D)  $6 x$ 
E)  $24 x$ 
A)  $16 \ln x$ 
B)  $4 - \ln x$ 
C)  $6 \ln x$ 
D)  $6 x$ 
E)  $24 x$

Question 5

Find the determinant of $$\left[ \begin{array} { c c c } 
0 & 1 & 1 \
2 & - 2 & 0 \
- 1 & - 8 & - 3
\end{array} \right]$$ .

Accepted Answer

A) 8 
B) -10 
C) -15 
D) -12 
E) 2 
A) 8 
B) -10 
C) -15 
D) -12 
E) 2

Question 6

Find all minors of the matrix.​ $$\left[ \begin{array} { c c c } 
- 9 & 0 & 6 \
4 & 5 & 8 \
8 & - 8 & 4
\end{array} \right]$$ ​

Accepted Answer

A)  $M _ { 11 } = 84 , M _ { 12 } = - 48 , M _ { 13 } = - 72 , M _ { 21 } = 48 , M _ { 22 } = - 84 , M _ { 23 } = 72 , M _ { 31 } = - 30 , M _ { 32 } = - 96 , M _ { 33 } = - 45$ 
B)  $M _ { 11 } = - 72 , M _ { 12 } = 48 , M _ { 13 } = - 84 , M _ { 21 } = 72 , M _ { 22 } = - 30 , M _ { 23 } = - 96 , M _ { 31 } = - 45 , M _ { 32 } = 84 , M _ { 33 } = - 48$ 
C)  $M _ { 11 } = 48 , M _ { 12 } = - 84 , M _ { 13 } = 72 , M _ { 21 } = - 30 , M _ { 22 } = - 96 , M _ { 23 } = - 45 , M _ { 31 } = 84 , M _ { 32 } = - 48 , M _ { 33 } = - 72$ 
D)  $M _ { 11 } = - 48 , M _ { 12 } = - 72 , M _ { 13 } = 48 , M _ { 21 } = - 84 , M _ { 22 } = 72 , M _ { 23 } = - 30 , M _ { 31 } = - 96 , M _ { 32 } = - 45 , M _ { 33 } = 84$ 
E)  $M _ { 11 } = - 30 , M _ { 12 } = - 96 , M _ { 13 } = - 45 , M _ { 21 } = 84 , M _ { 22 } = - 48 , M _ { 23 } = - 72 , M _ { 31 } = 48 , M _ { 32 } = - 84 , M _ { 33 } = 72$ 
A)  $M _ { 11 } = 84 , M _ { 12 } = - 48 , M _ { 13 } = - 72 , M _ { 21 } = 48 , M _ { 22 } = - 84 , M _ { 23 } = 72 , M _ { 31 } = - 30 , M _ { 32 } = - 96 , M _ { 33 } = - 45$ 
B)  $M _ { 11 } = - 72 , M _ { 12 } = 48 , M _ { 13 } = - 84 , M _ { 21 } = 72 , M _ { 22 } = - 30 , M _ { 23 } = - 96 , M _ { 31 } = - 45 , M _ { 32 } = 84 , M _ { 33 } = - 48$ 
C)  $M _ { 11 } = 48 , M _ { 12 } = - 84 , M _ { 13 } = 72 , M _ { 21 } = - 30 , M _ { 22 } = - 96 , M _ { 23 } = - 45 , M _ { 31 } = 84 , M _ { 32 } = - 48 , M _ { 33 } = - 72$ 
D)  $M _ { 11 } = - 48 , M _ { 12 } = - 72 , M _ { 13 } = 48 , M _ { 21 } = - 84 , M _ { 22 } = 72 , M _ { 23 } = - 30 , M _ { 31 } = - 96 , M _ { 32 } = - 45 , M _ { 33 } = 84$ 
E)  $M _ { 11 } = - 30 , M _ { 12 } = - 96 , M _ { 13 } = - 45 , M _ { 21 } = 84 , M _ { 22 } = - 48 , M _ { 23 } = - 72 , M _ { 31 } = 48 , M _ { 32 } = - 84 , M _ { 33 } = 72$

Question 7

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 
- \frac { 7 } { 3 } & \frac { 14 } { 3 } \\
- 1 & \frac { 1 } { 3 }
\end{array} \right]$$ ​

Accepted Answer

A)  $- \frac { 35 } { 17 }$ 
B)  $\frac { 13 } { 35 }$ 
C)  $\frac { 11 } { 35 }$ 
D)  $- \frac { 17 } { 35 }$ 
E)  $\frac { 35 } { 9 }$ 
A)  $- \frac { 35 } { 17 }$ 
B)  $\frac { 13 } { 35 }$ 
C)  $\frac { 11 } { 35 }$ 
D)  $- \frac { 17 } { 35 }$ 
E)  $\frac { 35 } { 9 }$

Question 8

Find all minors of the matrix.​ $$\left[ \begin{array} { c c c } 
- 7 & 0 & 4 \
2 & 4 & 3 \
3 & - 3 & 5
\end{array} \right]$$ ​

Accepted Answer

A)  $M _ { 11 } = 1 , M _ { 12 } = - 18 , M _ { 13 } = 12 , M _ { 21 } = - 47 , M _ { 22 } = 21 , M _ { 23 } = - 16 , M _ { 31 } = - 29 , M _ { 32 } = - 28 , M _ { 33 } = 29$ 
B)  $M _ { 11 } = 12 , M _ { 12 } = - 47 , M _ { 13 } = 21 , M _ { 21 } = - 16 , M _ { 22 } = - 29 , M _ { 23 } = - 28 , M _ { 31 } = 29 , M _ { 32 } = 1 , M _ { 33 } = - 18$ 
C)  $M _ { 11 } = 29 , M _ { 12 } = 1 , M _ { 13 } = - 18 , M _ { 21 } = 12 , M _ { 22 } = - 47 , M _ { 23 } = 21 , M _ { 31 } = - 16 , M _ { 32 } = - 29 , M _ { 33 } = - 28$ 
D)  $M _ { 11 } = - 18 , M _ { 12 } = 12 , M _ { 13 } = - 47 , M _ { 21 } = 21 , M _ { 22 } = - 16 , M _ { 23 } = - 29 , M _ { 31 } = - 28 , M _ { 32 } = 29 , M _ { 33 } = 1$ 
E)  $M _ { 11 } = - 16 , M _ { 12 } = - 29 , M _ { 13 } = - 28 , M _ { 21 } = 29 , M _ { 22 } = 1 , M _ { 23 } = - 18 , M _ { 31 } = 12 , M _ { 32 } = - 47 , M _ { 33 } = 21$ 
A)  $M _ { 11 } = 1 , M _ { 12 } = - 18 , M _ { 13 } = 12 , M _ { 21 } = - 47 , M _ { 22 } = 21 , M _ { 23 } = - 16 , M _ { 31 } = - 29 , M _ { 32 } = - 28 , M _ { 33 } = 29$ 
B)  $M _ { 11 } = 12 , M _ { 12 } = - 47 , M _ { 13 } = 21 , M _ { 21 } = - 16 , M _ { 22 } = - 29 , M _ { 23 } = - 28 , M _ { 31 } = 29 , M _ { 32 } = 1 , M _ { 33 } = - 18$ 
C)  $M _ { 11 } = 29 , M _ { 12 } = 1 , M _ { 13 } = - 18 , M _ { 21 } = 12 , M _ { 22 } = - 47 , M _ { 23 } = 21 , M _ { 31 } = - 16 , M _ { 32 } = - 29 , M _ { 33 } = - 28$ 
D)  $M _ { 11 } = - 18 , M _ { 12 } = 12 , M _ { 13 } = - 47 , M _ { 21 } = 21 , M _ { 22 } = - 16 , M _ { 23 } = - 29 , M _ { 31 } = - 28 , M _ { 32 } = 29 , M _ { 33 } = 1$ 
E)  $M _ { 11 } = - 16 , M _ { 12 } = - 29 , M _ { 13 } = - 28 , M _ { 21 } = 29 , M _ { 22 } = 1 , M _ { 23 } = - 18 , M _ { 31 } = 12 , M _ { 32 } = - 47 , M _ { 33 } = 21$

Question 9

Find the determinant of the matrix.​ $[ - 15 ]$ ​

Accepted Answer

A)  $15$ 
B)  $- \frac { 1 } { 15 }$ 
C)  $- 15$ 
D) 0 
E)  $\frac { 1 } { 15 }$ 
A)  $15$ 
B)  $- \frac { 1 } { 15 }$ 
C)  $- 15$ 
D) 0 
E)  $\frac { 1 } { 15 }$

Question 10

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 
- 5 & 2 \
\frac { 1 } { 2 } & 1
\end{array} \right]$$ ​

Accepted Answer

A) -6 
B) -7 
C) -4 
D) -5 
E) -8 
A) -6 
B) -7 
C) -4 
D) -5 
E) -8

Question 11

Find the determinant of $$\left[ \begin{array} { c c c } 
0 & - 2 & 0 \
- 3 & 2 & - 6 \
1 & 0 & 6
\end{array} \right]$$ by the method of expansion by cofactors using column 3. ​

Accepted Answer

A) -14 
B) 24 
C) -4 
D) -24 
E) -34 
A) -14 
B) 24 
C) -4 
D) -24 
E) -34

Question 12

Find the determinant of the matrix.​ $$\left[ \begin{array} { c c } 
- \frac { 1 } { 2 } & \frac { 1 } { 3 } \\
- 24 & \frac { 1 } { 3 }
\end{array} \right]$$ ​

Accepted Answer

A)  $\frac { 47 } { 6 }$ 
B)  $- \frac { 6 } { 47 }$ 
C)  $- \frac { 47 } { 6 }$ 
D)  $\frac { 3 } { 47 }$ 
E)  $\frac { 6 } { 47 }$ 
A)  $\frac { 47 } { 6 }$ 
B)  $- \frac { 6 } { 47 }$ 
C)  $- \frac { 47 } { 6 }$ 
D)  $\frac { 3 } { 47 }$ 
E)  $\frac { 6 } { 47 }$

Exam 50: The Determinant of a Square Matrix

Find the determinant of the matrix.​ [−4058]\left[ \begin{array} { c c } - 4 & 0 \\5 & 8\end{array} \right][−45​08​] ​

Evaluate the determinant.​ ∣55−4241−41−121−412−5−1∣\left| \begin{array} { c c c c } 5 & 5 & - 4 & 2 \\4 & 1 & - 4 & 1 \\- 1 & 2 & 1 & - 4 \\1 & 2 & - 5 & - 1\end{array} \right|​54−11​5122​−4−41−5​21−4−1​​

Evaluate the determinant ∣4xxln⁡x46+ln⁡x∣\left| \begin{array} { c c } 4 x & x \ln x \\4 & 6 + \ln x\end{array} \right|​4x4​xlnx6+lnx​​ in which the entries are functions.

Find the determinant of [0112−20−1−8−3]\left[ \begin{array} { c c c } 0 & 1 & 1 \\2 & - 2 & 0 \\- 1 & - 8 & - 3\end{array} \right]​02−1​1−2−8​10−3​​ .

Find all minors of the matrix.​ [−9064588−84]\left[ \begin{array} { c c c } - 9 & 0 & 6 \\4 & 5 & 8 \\8 & - 8 & 4\end{array} \right]​−948​05−8​684​​ ​

Find the determinant of the matrix.​ [−73143−113]\left[ \begin{array} { c c } - \frac { 7 } { 3 } & \frac { 14 } { 3 } \\\\- 1 & \frac { 1 } { 3 }\end{array} \right]​−37​−1​314​31​​​ ​

Find all minors of the matrix.​ [−7042433−35]\left[ \begin{array} { c c c } - 7 & 0 & 4 \\2 & 4 & 3 \\3 & - 3 & 5\end{array} \right]​−723​04−3​435​​ ​

Find the determinant of the matrix.​ [−15][ - 15 ][−15] ​

Find the determinant of the matrix.​ [−52121]\left[ \begin{array} { c c } - 5 & 2 \\\frac { 1 } { 2 } & 1\end{array} \right][−521​​21​] ​

Find the determinant of [0−20−32−6106]\left[ \begin{array} { c c c } 0 & - 2 & 0 \\- 3 & 2 & - 6 \\1 & 0 & 6\end{array} \right]​0−31​−220​0−66​​ by the method of expansion by cofactors using column 3. ​

Find the determinant of the matrix.​ [−1213−2413]\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \\\\- 24 & \frac { 1 } { 3 }\end{array} \right]​−21​−24​31​31​​​ ​

Rectangular Coordinates

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A Library of Parent Functions

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Mathematical Modeling and Variation

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Complex Numbers

Zeros of Polynomial Functions

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Nonlinear Inequalities

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Properties of Logarithms

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Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

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Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

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Linear Programming

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The Inverse of a Square Matrix

Applications of Matrices and Determinants

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Mathematical Induction

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Introduction to Conics Parabolas

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Polar Equations of Conics

Graphs of Polar Equations

Filters

Find the determinant of the matrix. $\left[ \begin{array} { c c } - 4 & 0 \\5 & 8\end{array} \right]$

Evaluate the determinant. $\left| \begin{array} { c c c c } 5 & 5 & - 4 & 2 \\4 & 1 & - 4 & 1 \\- 1 & 2 & 1 & - 4 \\1 & 2 & - 5 & - 1\end{array} \right|$

Evaluate the determinant $\left| \begin{array} { c c } 4 x & x \ln x \\4 & 6 + \ln x\end{array} \right|$ in which the entries are functions.

Find the determinant of $\left[ \begin{array} { c c c } 0 & 1 & 1 \\2 & - 2 & 0 \\- 1 & - 8 & - 3\end{array} \right]$ .

Find all minors of the matrix. $\left[ \begin{array} { c c c } - 9 & 0 & 6 \\4 & 5 & 8 \\8 & - 8 & 4\end{array} \right]$

Find the determinant of the matrix. $\left[ \begin{array} { c c } - \frac { 7 } { 3 } & \frac { 14 } { 3 } \\\\- 1 & \frac { 1 } { 3 }\end{array} \right]$

Find all minors of the matrix. $\left[ \begin{array} { c c c } - 7 & 0 & 4 \\2 & 4 & 3 \\3 & - 3 & 5\end{array} \right]$

Find the determinant of the matrix. $[ - 15 ]$

Find the determinant of the matrix. $\left[ \begin{array} { c c } - 5 & 2 \\\frac { 1 } { 2 } & 1\end{array} \right]$

Find the determinant of $\left[ \begin{array} { c c c } 0 & - 2 & 0 \\- 3 & 2 & - 6 \\1 & 0 & 6\end{array} \right]$ by the method of expansion by cofactors using column 3.

Find the determinant of the matrix. $\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \\\\- 24 & \frac { 1 } { 3 }\end{array} \right]$