Determinant of 4x4 Matrix: Steps, Formula & Solved Examples

Determinants are the scalars that are obtained by the sum of the product of elements of a square matrix and their cofactors by following a particular rule. Determinants in maths help us in finding the adjoint of a matrix and also the inverse of a matrix. We can use determinants to solve the linear equations through the matrix inversion method. Also, determinants are helpful in finding the cross-product of two vectors.

What is Determinant of 4×4 Matrix?

Determinant of a 4x4 matrix is a unique number that is calculated using a special formula. If a matrix is of order n x n then it is a square matrix. So, here 4x4 is a square matrix having 4 rows and 4 columns. Also for a square matrix A that is of the order 4 × 4, its determinant is written as lAl.

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There can be different ways to find the determinants of a matrix of an order 4 × 4.

Before trying to calculate the value of the Determinant of 4×4 matrix, we first need to check if the value of the determinant is a non-zero number.

This can be done by simply analyzing the determinant provided.

Some of the cases where the value of determinant of 4×4 matrix is zero are:

Case 1:

In this case, one of the rows in the matrix is zero, so the value of the determinant for the matrix becomes zero.

Case 2:

In the above matrix, as we can see that the first and the third column have the same values, so the value for the Determinant of the Matrix is zero.

Case 3:

In the above scenario, we can see that the second and the third row are proportional to each other. So, the value of the determinant for Matrix is zero.

Now that we know the cases where the value of the determinant can be zero, let us check the ways to find the value of the determinant for Matrix in case it is non-zero.

What is Gaussian Elimination?

Gaussian Elimination method is also termed as row reduction method and is used for solving linear equations in linear algebra. This elimination method is actually a series of operations that are used on a matrix and is very helpful in finding the rank of a matrix, inverse of a matrix, and even determinant of a matrix.

In order to perform this elimination we use a series of algebraic operations on a matrix making all the elements below the main diagonal as zero.

We can use three kinds of operations on rows:

• By interchanging two rows: The value of the determinant changes its sign. Negative becomes positive and positive becomes negative.
• The value of the determinant remains unchanged on multiplying the row with a non-zero constant.
• The value of the determinant remains unaffected by adding or subtracting a row from any other row.

Learn about Involutory Matrix

Triangular Property of a Matrix

In order to find the determinant matrix we must be well acquainted with the triangular property of a matrix. By the triangular property of a matrix, we say that if every element of the matrix above as well as below the main diagonal is zero, then the value of the determinant is equal to the product of diagonal elements.

Now, there are three kinds of triangular matrices, we will learn about each one of these below:

1. Upper Triangular Matrix: A Matrix is called an upper triangular matrix when all the elements below the main diagonal are zero. Or we can say that all the non-zero elements of such a matrix are above the main diagonal.

Example:

is an upper triangular matrix.

2. Lower Triangular Matrix: A Matrix is called a lower triangular matrix when all the elements above the main diagonal are zero. Or we can say that all the non-zero elements of such a matrix are below the main diagonal.

Example:

is a lower triangular matrix.

3. Diagonal Matrix: A Matrix is called a diagonal matrix when all the elements except the main diagonal are zero. Or we can say that in such a matrix everything above or below the main diagonal is zero.

Example:

The value of the determinant in all the above three cases is the product of diagonal elements. Let us understand this using an example:

Example: Let the given

We know that adding rows and columns to other rows and columns of the same determinant does not cause any change to the value of the determinant.

Using the operations:

The resultant matrix will look like this:

As we know that for a triangular matrix, the determinant is equal to the product of diagonal elements.

So, |B| = 2.2.2.5 = 40.

Learn about Applications of Matrices and Determinants

What is the Cofactor Expansion Method?

In order to find the determinant for a Matrix we use another method which is cofactor expansion. Here is how to perform this method.

In a Matrix select any row or column and we multiply each element of the row or column with their corresponding cofactors. And finally, we find the sum of all the products.

It is therefore recommended to select the row or column with the maximum number of zeroes. This will reduce the calculation time and also the chances of error.

Let us understand this using an example:

In this matrix, we can see that row 4 has three zero entries. So, let us use the cofactor expansion method along row 4.

Let us begin with the first element of row 4, i.e. 0. We multiply zero with , where m is the row number and n is the column number. We further multiply it with determinant obtained after eliminating the row and the column of the corresponding element.

So, we get

In the same way, we move on to the rest of the elements of row 4 and get,

We will now add all these statements.

= +++

= 0 + 0 + + 0

=(-5)(-1)

=5

Again we expand determinant as row 3 has 2 zeroes so we use cofactor expansion on row 3.

=\(5\left(\left(0\right)\left(-1\right)^{3+1}\times\left|\begin{matrix}0&-2\\
0&-5\end{matrix}\right|+\left(1\right)\left(-1\right)^{3+2}\times\left|\begin{matrix}-1&-2\\
1&-5\end{matrix}\right|+\left(0\right)\left(-1\right)^{3+3}\times\left|\begin{matrix}-1&0\\
1&0\end{matrix}\right|\right)\)

=\(5\left(0+\left(1\right)\left(-1\right)^5\times\left|\begin{matrix}-1&-2\\
1&-5\end{matrix}\right|+0\right)\)

=\(5\times -1\times \left|\begin{matrix}-1&-2\\
1&-5\end{matrix}\right|\)

= (5)(-7)= -35

Learn about Cofactor Matrix

If you are checking Determinant of 4 x 4 Matrix article, also check related maths articles:
Involutory Matrix	Minor of a Matrix
Inverse of Matrix	Matrix Operations
Types of Matrices	Cofactor Matrix

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