# Determinant of a 2×2 matrix

On this post we explain you what the determinant of a 2×2 matrix is and how to find the determinant of a 2×2 matrix. In addition, you will find examples and exercises solved step by step on how to solve determinants of order 2, so that you can practice and understand perfectly how to do it.

## What is a 2×2 determinant?

A determinant of order 2 is a 2×2 dimension matrix represented with a vertical bar on each side of the matrix. For example, if we have the following matrix:

The determinant of matrix A is represented as follows:

As you have seen, writing the determinant of a 2×2 square matrix is easy. Now let’s see how to calculate the determinant of a 2×2 matrix with its formula.

## How to find the determinant of a 2×2 matrix?

To calculate the determinant of a 2×2 matrix, multiply the elements of the 2×2 matrix on the main diagonal and subtract the product of the elements on the secondary diagonal.

## Examples of determinants of 2×2 matrices:

You can see the process to calculate 2×2 determinants in the following examples:

Note that the determinant of a matrix can only be calculated if the matrix is square.

Now that you have seen how to compute the determinant of a 2×2 matrix, you should see how to find the determinant of a 3×3 matrix.

## Practice determinants of 2×2 matrices

### Problem 1

Calculate the determinant of the following 2×2 matrix:

To solve the determinant of a 2×2 matrix we have to multiply the elements on the main diagonal and subtract the product of the secondary diagonal:

### Problem 2

Solve the determinant of the following 2×2 dimension matrix:

To find the solution of a determinant of order 2 we must multiply the elements of the matrix on the main diagonal and subtract the product of the secondary diagonal:

### Problem 3

Find the determinant of the following 2×2 matrix:

To find the determinant of a 2×2 square matrix, we must first multiply the elements on the main diagonal and then subtract the product of the secondary diagonal:

### Problem 4

Take the determinant of the following 2×2 matrix:

To calculate determinants of 2×2 matrices we have to multiply the elements on the main diagonal and subtract the product of the secondary diagonal, so:

### Problem 5

Determine the result of the following 2×2 determinant:

To find the solution of a 2-by-2 determinant we simply have to multiply the elements on the main diagonal and subtract the product of the secondary diagonal:

### Problem 6

Find the value of x in the following 2×2 matrix if its determinant equals to 8:

First we get the determinant of the matrix by applying the formula:

The problem says that the determinant results in 8, thus, we have to set the expression obtained equal to 8:

And we solve the linear equation to find the value of x:

Did you know that 2×2 determinants are used to solve systems of linear equations? To do this, we use Cramer’s rule 2×2.

## Can the determinant of a 2×2 matrix be zero?

The determinant of a 2×2 matrix can be equal to zero, for example:

However, the result of the determinant indicates the invertibility of the matrix:

Therefore, it is very important whether the determinant of a matrix of order 2 is 0 or not, since it will be a type of matrix or another.

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