How to calculate the determinant of a matrix

The determinant of a square matrix is calculated as a scalar value that is used to determine the different properties of a matrix.

For instance, the determinant is used in calculating:

• The inverse of a matrix

• Eigenvalues and eigenvectors

• Solving differential equations

• Solving a system of linear equations

• Analyzing the solution of a system of equations

Method

Numerous procedures can be implemented to calculate the determinant of a matrix. Two of the methods are listed below:

• Gauss-Jordan elimination

• Laplace expansion (Cofactor method)

Gauss-Jordan elimination

The Gauss-Jordan elimination method is an easy and effective procedure for calculating the determinant of an $n \times n$ matrix.

Follow the steps below:

• Convert the matrix into an upper triangularAll entries in the matrix are above the diagonal., lower triangularAll entries in the matrix are below the diagonal., or diagonal matrixThe matrix contains entries on the diagonal only..

• The determinant equates to the multiplication of all the diagonal entries.

Example

Calculate the determinant of the following matrix:

Using the Gauss-Jordan elimination procedure, convert the matrix into an upper triangular matrix:

The determinant of the matrix $A$ equates to:

Laplace expansion

Laplace expansion—also known as the cofactor expansion method—is an efficient procedure to calculate the determinant of a matrix.

The following equation is used in the calculations:

Where:

• $C_{ij}$ represents the cofactor at the $ij$ position.

• $M_{ij}$ represents a minor - determinant of smaller matrices.

For instance, the cofactor $C_{11}$ for the matrix given above can be calculated as follows:

The determinant of a matrix can be calculated by choosing any row or column and summing up its cofactors.

Example

Laplace expansion has been used over the third row to calculate the determinant of the matrix.

Quiz

Here are a few questions with solutions that you can practice and enhance your skill set: