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Linear Algebra : The Determinant

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Example Questions

Example Question #1 : The Determinant

Calculate the determinant of matrix A where

$A = \begin{bmatrix} 5&4 \\ 1&2 \\ 0&7 \end{bmatrix}$ $\displaystyle A = \begin{bmatrix} 5&4 \\ 1&2 \\ 0&7 \end{bmatrix}$

Possible Answers:

Not Possible

-50

Correct answer:

Not Possible

Explanation:

The matrix must be square to calculate its determinant, therefore, it is not possible to calculate the determinant for this matrix.

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Example Question #2 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 4&1 \\ 5& 3 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} 4&1 \\ 5& 3 \end{bmatrix}$

Possible Answers:

-7

Correct answer:

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$ $\displaystyle a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

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Example Question #3 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 8&-8 \\ 51&12 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} 8&-8 \\ 51&12 \end{bmatrix}$

Possible Answers:

-504

504

-315

Correct answer:

504

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$ $\displaystyle a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

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Example Question #4 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 8&6 \\ 0&2 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} 8&6 \\ 0&2 \end{bmatrix}$

Possible Answers:

-15

Correct answer:

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$ $\displaystyle a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

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Example Question #5 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 2&5 &6 \\ 2&1 &0 \\ 5&3 &4 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} 2&5 &6 \\ 2&1 &0 \\ 5&3 &4 \end{bmatrix}$

Possible Answers:

-26

-24

Correct answer:

-26

Explanation:

Calculating the determinant of a 3x3 matrix is more difficult than a 2x2 matrix. To calculate the determinant of a 3x3 matrix, we use the following $\left | A\right |=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31})$ $\displaystyle \left | A\right |=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31})$

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Example Question #6 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} -5&7 &10 \\ -20&15 &2 \\ 0& 3& 8 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} -5&7 &10 \\ -20&15 &2 \\ 0& 3& 8 \end{bmatrix}$

Possible Answers:

-49

-50

Correct answer:

-50

Explanation:

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Example Question #7 : The Determinant

Calculate the determinant of $\begin{bmatrix} 5&-2 \\ 1&3 \end{bmatrix}$ $\displaystyle \begin{bmatrix} 5&-2 \\ 1&3 \end{bmatrix}$.

Possible Answers:

$\displaystyle 6$

$\displaystyle 17$

$\displaystyle -1$

$\displaystyle 13$

Correct answer:

$\displaystyle 17$

Explanation:

By definition,

$\begin{vmatrix} a&b \\ c&d \end{vmatrix}= a(d)-b(c)$ $\displaystyle \begin{vmatrix} a&b \\ c&d \end{vmatrix}= a(d)-b(c)$,

therefore,

$\begin{vmatrix} 5&-2 \\ 1&3 \end{vmatrix}= 5(3)-1(-2)= 17$ $\displaystyle \begin{vmatrix} 5&-2 \\ 1&3 \end{vmatrix}= 5(3)-1(-2)= 17$.

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Example Question #8 : The Determinant

Calculate the determinant of $\begin{bmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{bmatrix}$ $\displaystyle \begin{bmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{bmatrix}$

Possible Answers:

$\displaystyle 0$

$\displaystyle 21$

$\displaystyle -4$

$\displaystyle 4$

Correct answer:

$\displaystyle -4$

Explanation:

For simplicity, we will find the determinant by expanding along the second row. Consider the following:

$\begin{vmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{vmatrix}= -0[3(2)-1(8)]+2[1(2)-1(4)]-0[1(8)-3(4)]= -4$ $\displaystyle \begin{vmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{vmatrix}= -0[3(2)-1(8)]+2[1(2)-1(4)]-0[1(8)-3(4)]= -4$

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Example Question #8 : The Determinant

Calculate the determinant of $\begin{bmatrix} 1&2 \\ 3&6 \end{bmatrix}$ $\displaystyle \begin{bmatrix} 1&2 \\ 3&6 \end{bmatrix}$.

Possible Answers:

$\displaystyle 16$

$\displaystyle 0$

$\displaystyle 12$

$\displaystyle -4$

Correct answer:

$\displaystyle 0$

Explanation:

By definition,

$\begin{vmatrix} a&b \\ c&d \end{vmatrix}= ad-bc \Rightarrow \begin{vmatrix} 1&2 \\ 3&6 \end{vmatrix} = 1(6)- 2(3) = 0$ $\displaystyle \begin{vmatrix} a&b \\ c&d \end{vmatrix}= ad-bc \Rightarrow \begin{vmatrix} 1&2 \\ 3&6 \end{vmatrix} = 1(6)- 2(3) = 0$.

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Example Question #7 : The Determinant

Calculate the determinant of matrix A.

$A =\begin{bmatrix} 5&3 \\ 6&4 \end{bmatrix}$ $\displaystyle A =\begin{bmatrix} 5&3 \\ 6&4 \end{bmatrix}$

Possible Answers:

$\displaystyle -2$

$\displaystyle 38$

$\displaystyle 2$

$\displaystyle 1$

Not possible

Correct answer:

$\displaystyle 2$

Explanation:

In order to find the determinant of a 2x2 matrix, compute $det A= a_{11}a_{22}-a_{12}a_{21}$ $\displaystyle det A= a_{11}a_{22}-a_{12}a_{21}$ :

$\displaystyle = 5(4)-3(6)$

=20-18 $\displaystyle =20-18$

$\displaystyle =2$

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Linear Algebra : The Determinant

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : The Determinant

Example Question #2 : The Determinant

Example Question #3 : The Determinant

Example Question #4 : The Determinant

Example Question #5 : The Determinant

Example Question #6 : The Determinant

Example Question #7 : The Determinant

Example Question #8 : The Determinant

Example Question #8 : The Determinant

Example Question #7 : The Determinant

All Linear Algebra Resources

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