How to Find Determinants of a Matrix

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What is the Determinant of a Matrix?

The determinant of a matrix is a unique number that is only defined for square matrices (matrices with the same number of rows and columns). A determinant is used in calculus and other algebra-related matrices. It is a real number representing the matrix and can be used to solve a system of linear equations and find the inverse of a matrix.

How to find the Determinant of a Matrix?

The following steps can be used to find the value of a matrix's determinant:

Get the cofactor for each element in the first row or column.
Multiply each element by the determinant of the corresponding cofactor.
Add the results with the signs swapped.

As a starting point, the value of a matrix's determinant of a $1 \times 1$ matrix is the single value itself.
The cofactor of an element is a matrix that can be made by taking that element's row and column out of that Matrix.

Determinant of 2 × 2 Matrix:

$\left[ \begin{matrix} a & b\\ c & d\\ \end{matrix} \right]$ $|A|=a\ d\ -\ b\ c$

Determinant of 3 × 3 Matrix:

$\left[ \begin{matrix} a & b &c \\ d & e & f \\ g & h & i \\ \end{matrix} \right]$ $|A|=ad-bc= a\ (e\ i \ –\ f\ h) \ –\ b\ (d\ i\ -\ f\ g) \ +\ c\ (d\ h \ –\ e\ g)$

Properties of Determinants of a Matrix

● A determinant can only be used with a square matrix ( $1 \times 1$ , $2 \times 2$ , $3 \times 3$ , $4 \times 4$ ,...).
● A determinant can be a real or complex number.
● $|A|$ doesn't show the modulus of $A$ in this case; it shows the determinant of matrix $A$ .
● If the $2 \times 2$ matrix's elements are all the same, the determinant will be $0$ .
● If every number in a row or column of a $2 \times 2$ matrix is $0$ , then the determinant is also $0$ .
● The determinant of the product of two matrices is the same as the product of their determinants. $|AB| \ = \ |A| \ |B|$ .

Free printable Worksheets

Exercises for Finding Determinants of a Matrix

1) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 4 & -8 \\\ -2 & 3 \end{bmatrix}$

2) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 7 & 3 \\\ 1 & 6 \end{bmatrix}$

3) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 5 & 4 \\\ 2 & -9 \end{bmatrix}$

4) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 0 & -9 \\\ -4 & 1 \end{bmatrix}$

5) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 5 & 7 \\\ 3 & 2 \end{bmatrix}$

6) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 9 & 8 \\\ 11 & 7 \end{bmatrix}$

7) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 12 & 6 \\\ -4 & 7 \end{bmatrix}$

8) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 6 & 11 \\\ 9 & 13 \end{bmatrix}$

9) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} -5 & 17 \\\ 3 & -4 \end{bmatrix}$

10) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 4 & -1 & 8 \\\ 1 & 5 & -2 \\\ -3 & 4 & 6 \end{bmatrix}$

Answers

1) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 4 & -8 \\\ -2 & 3 \end{bmatrix}$

$\color{red}{|A| \ = \ 4 \times 3 \ - \ (-8) \times (-2) \ = \ 12 \ - \ 16 \ = \ -4}$

2) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 7 & 3 \\\ 1 & 6 \end{bmatrix}$

$\color{red}{|A| \ = \ 7 \times 6 \ - \ 3 \times 1 \ = \ 42 \ - \ 3 \ = \ 39}$

3) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 5 & 4 \\\ 2 & -9 \end{bmatrix}$

$\color{red}{|A| \ = \ -2 \times (-3) \ - \ 6 \times 5 \ = \ 6 \ - \ 30 \ = \ -24}$

4) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 0 & -9 \\\ -4 & 1 \end{bmatrix}$

$\color{red}{|A| \ = \ 0 \times 1 \ - \ (-9) \times (-4) \ = \ -36}$

5) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 5 & 7 \\\ 3 & 2 \end{bmatrix}$

$\color{red}{|A| \ = \ 5 \times 2 \ - \ 7 \times 3 \ = \ 10 \ - \ 21 \ = \ -11}$

6) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 9 & 8 \\\ 11 & 7 \end{bmatrix}$

$\color{red}{|A| \ = \ 9 \times 7 \ - \ 8 \times 11 \ = \ 63 \ - \ 88 \ = \ -25}$

7) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 12 & 6 \\\ -4 & 7 \end{bmatrix}$

$\color{red}{|A| \ = \ 12 \times 7 \ - \ 6 \times (-4) \ = \ 84 \ - \ (-24) \ = \ 108}$

8) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 6 & 11 \\\ 9 & 13 \end{bmatrix}$

$\color{red}{|A| \ = \ 6 \times 13 \ - \ 11 \times 9 \ = \ 78 \ - \ 99 \ = \ -21}$

9) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} -5 & 17 \\\ 3 & -4 \end{bmatrix}$

$\color{red}{|A| \ = \ -5 \times (-4) \ - \ 17 \times 3 \ = \ 20 \ - \ 51 \ = \ -31}$

10) Find the determinant of the matrix: $A \ = \ \begin{bmatrix} 4 & -1 & 8 \\\ 1 & 5 & -2 \\\ -3 & 4 & 6 \end{bmatrix}$

$\color{red}{det \begin{bmatrix} 5 & -2 \\\ 4 & 6 \end{bmatrix} \ = \ 38}$
$\color{red}{det \begin{bmatrix} 1 & -2 \\\ -3 & 6 \end{bmatrix} \ = \ 0}$
$\color{red}{det \begin{bmatrix} 1 & 5 \\\ -3 & 4 \end{bmatrix} \ = \ 19}$
$\color{red}{|A| \ = \ 4 \times 38 \ - \ (-1) \times 0 \ + \ 8 \times 19 \ = \ 304}$

How to Find Determinants of a Matrix