# Determinant of a Matrix – Explanation & Examples

The determinant of a matrix is a scalar value of immense importance. With the help of the determinant of matrices, we can find useful information of linear systems, solve linear systems, find the inverse of a matrix, and use it in calculus. Let’s take a look at the definition of the determinant:The determinant of a matrix is a scalar value that results from certain operations with the elements of the matrix.In this lesson, we will look at the determinant, how to find the determinant, the formula for the determinant of $2 \times 2$ and $3 \times 3$ matrices, and examples to clarify our understanding of determinants. Let us start!

## What is the Determinant of a Matrix?

The determinant of a matrix is a single constant value (or, a scalar value) that tells us certain things about the matrix. The value of the determinant results from certain operations that we do with the elements of a matrix.There are $3$ ways we use to denote the determinant of a matrix. Check the picture below:On the left-hand side is Matrix $A$. This is how we write a matrix.On the right-hand side are $3$ notations for determinants of matrices. We can denote the determinant of Matrix $A$ by writing $det( A )$, $| A |$, or by putting all the elements of the matrix inside two vertical bars (as shown). All these $3$ notations denote the determinant of a matrix.

## How to Find the Determinant of a Matrix

So how do we find the determinant of matrices?First of all, we can only calculate the determinant for square matrices!There aren’t any determinant for non-square matrices.Now, there is a formula (algorithm) to find the determinant of any square matrix. That is out of the scope of this lesson. Rather, we will look at finding determinants of $2 \times 2$ matrices and $3 \times 3$ matrices. The formula can be extended to find the determinant of $4 \times 4$ matrices, but that is too complicated and messy!Below, we look at the formula for $2 \times 2$ matrices and $3 \times 3$ matrices and see how to calculate the determinant of such matrices.

## Matrix Determinant Formula

We will find the determinant of $2 \times 2$ and $3 \times 3$ matrices in this section.

### Practice Questions

1. Find the determinant of the matrix shown below: $A = \begin{bmatrix} – 5 & – 10 \\ 3 & – 1 \end{bmatrix}$
2. Find $y$ given $\begin{vmatrix} { 1 } & { 3 } & { – 1 } \\ { 5 } & { 0 } & { y } \\ { – 1 } & { 2 } & { 3 } \end {vmatrix} = – 60$

1. Matrix $A$, a $2 \times 2$ matrix, is given. We need to find the determinant of it. We do so by applying the formula. Process is shown below:$det( A ) = | A | = \begin{vmatrix} { – 5 } & { – 10 } \\ { 3 } & { – 1 } \end {vmatrix} $$= ( – 5 ) ( – 1 ) – ( – 10 ) ( 3 )$$ = 5 + 30$$= 35$
2. We are already given the determinant and have to find an element, $y$. Let’s put it into the formula for the determinant of a $3 \times 3$ matrix and solve for $y$:$\begin{vmatrix} { 1 } & { 3 } & { – 1 } \\ { 5 } & { 0 } & { y } \\ { – 1 } & { 2 } & { 3 } \end {vmatrix} = – 60$ $1 [ ( 0 )( 3 ) – ( y )( 2 ) ] – 3 [ ( 5 )( 3 ) – ( y )( – 1 ) ] + (-1) [ ( 5 )( 2 ) – ( 0 )( – 1 ) ] = – 60$ $1 [- 2y ] – 3 [ 15 + y ] + (-1) [ 10 ] = – 60$ $– 2y – 45 – 3y – 10 = – 60$ $– 5y – 55 = – 60$ $– 5y = – 60 + 55$ $– 5y = – 5$ $y = 1$