# Determinant of a 4×4 matrix

We explain how to solve the determinant of a 4×4 matrix with an example. Also, you will find solved exercises so that you can practice and perfectly understand how to compute the determinant of a 4×4 matrix.

## How to find the determinant of a 4×4 matrix using row operations and cofactor expansion

Let’s see how to compute the determinant of a 4×4 matrix solving an example:

The first step in computing the determinant of a 4×4 matrix is to make zero all the elements of a column except one using elementary row operations.

We can perform elementary row operations thanks to the properties of determinants.

In this case, the first column already has a zero. Thus, we are going to transform all the entries in the first column to 0 except for the number 1 (since it is easier to do calculations with the row that has a 1). To do so, we add the first row to the second row, and we subtract the first row multiplied by 2 from the fourth row:

Once we have transformed to 0 all the elements except one of the chosen column, we compute the determinant of the 4×4 matrix using cofactor expansion.

The cofactor expansion is a method to find determinants which consists in adding the products of the elements of a column by their respective cofactors.

Being the i, j cofactor of the matrix defined by:

Where Mij is the i, j minor of the matrix, that is, the determinant that results from deleting the i-th row and the j-th column of the matrix.

So, we add the products of the elements in the first column by their respective cofactors:

The terms multiplied by 0 cancel out, so we simplify them:

So we have to compute the cofactor of the first row and the first column, that is, (-1) raised to 1+1 (1st row and 1st column) multiplied by the determinant that results from deleting the first row and the first column of the 4×4 matrix.

And, finally, we simply have to find the determinant of a 3×3 matrix:

Thus, we have solved the determinant of a 4×4 matrix using row operations and the cofactor expansion.

## Practice problems on finding the determinant of a 4×4 matrix

### Problem 1

Find the determinant of the following square matrix of order 4:

We will find the determinant of the 4×4 matrix with the cofactor expansion method, also called Laplace expansion. But first we do elementary operations with the rows in order to transform to zero all the elements of a column except one :

And now we compute the 4×4 determinant using the cofactor expansion:

We simplify the terms:

And finally we perform the cofactor:

### Problem 2

Compute the determinant of the following 4×4 matrix:

We will calculate the determinant of the 4×4 determinant by the cofactor expansion formula. But we first do elementary operations with the rows to convert all the elements of a column except one to zero:

Now we solve the determinant of the 4×4 by cofactor expansion:

We simplify the terms:

And we compute the cofactor of the third row and the second column:

### Problem 3

Evaluate the determinant of the following 4×4 matrix:

First of all, we will simplify the determinant of the 4×4 using row operations. The objetive is to make zeroes all the entries of a column except one:

Now we solve the 4×4 determinant using cofactor expansion:

We simplify all the cofactors multiplied by 0:

And we find the 4×4 determinant by solving the cofactor:

### Problem 4

Calculate the determinant of the following 4×4 dimension matrix:

We will calculate the determinant 4×4 by the Laplace’s rule. But first we must do operations with the rows to make zero all the elements of a column except one:

Now we find the 4-by-4 determinant by using the cofactors expansion method:

We solve the products:

And we find the cofactor from the first column and the fourth row:

### 2 thoughts on “Determinant of a 4×4 matrix”

1. Chase McKinley

Thanks so much for this! This made determinants substantially less tedious to take for 4×4 matrices. Excellent practice problems/explanations!

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