# How to Find Determinants of a Matrix?

For every square matrix, you can calculate the determinant of the matrix. Here is a step-by-step guide to finding the determinants of a matrix.

A Matrix is an array of numbers: $$m×n$$ with $$m$$ rows and $$n$$ columns. The determinant of a matrix is a scalar value that is defined for square matrices.

## A step-by-step guide to finding determinants of a matrix

• The determinant of a $$2×2$$ matrix $$A$$, $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$ is defined as $$|A|=ad-bc$$.
• The determinant of a $$3×3$$ matrix $$A$$, $$A=\begin{bmatrix}a & b & c \\d & e & f \\g & h & i\end{bmatrix}$$ is defined as $$|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$$

### Finding Determinants of a Matrix– Example 1:

Evaluate the determinant of matrix $$A=\begin{bmatrix}5 & -1 \\6 & 2 \end{bmatrix}$$

Solution:

The determinant is: $$|A|=5(2)-(-1)(6)=10-(-6)=10+6=16$$

### Finding Determinants of a Matrix– Example 2:

Evaluate the determinant of matrix: $$A=\begin{bmatrix}2 & 0 & 1 \\0 & -1 & 1 \\3 & 1 & -2\end{bmatrix}$$

Solution:

The determinant is: $$|A|=2((-1)(-2)-(1)(1))-0((-2)(0)-(1)(3)+1((0)(1)-(-1)(3))=5$$

## Exercises for Finding Determinants of a matrix

### Evaluate the determinants of each matrix.

• $$\color{blue}{\begin{bmatrix}3 & 5 \\0 & 9 \end{bmatrix}}$$
• $$\color{blue}{\begin{bmatrix}0 & 1 \\4 & 6 \end{bmatrix}}$$
• $$\color{blue}{\begin{bmatrix}1 & 5 & 4\\0 & 9 & 1\\1 & 0 & 6\end{bmatrix}}$$
• $$\color{blue}{\begin{bmatrix}6 & 0 & 5\\1 & 4 & 2\\3 & 7 & 4\end{bmatrix}}$$
• $$\color{blue}{27}$$
• $$\color{blue}{-4}$$
• $$\color{blue}{23}$$
• $$\color{blue}{-13}$$

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