How to calculate the inverse of a matrix

Overview

A square matrix $A$ is said to be invertible if there is another matrix, $C$ such that, $AC = I$ or $CA = I$, where $I$ represents the identity matrix. Matrix $C$ is said to be the inverse of matrix $A$ and is denoted as $A^{-1}$:

Properties

An invertible matrix $A$ should fulfill the following identities:

• $A(A^{-1}) = (A^{-1})A = I$

• $(A^{-1})^{-1} = A$

• $(A^T)^{-1} = (A^{-1})^T$

• $(AB)^{-1} = B^{-1}A^{-1}$

• $(A^k)^{-1} = (A^{-1})^k$, where $k$ is a constant power of the matrix.

The invertible matrix $A$ must also maintain the following properties:

• Matrix $A$ should be a square matrix with $n \times n$ dimensions.

• The rows and columns of matrix $A$ must be linearly independent.

• Any diagonal entry of matrix $A$ must not be zero.

• The solution to $Ax = b$ can be calculated as $x= A^{-1}b$ if matrix $A$ is invertible.

• A square matrix $A$ with orthonormal columns follows the identity $A^TA = I$. Hence, its inverse will be $A^{-1} = A^T$.

The inverse of a $2 \times 2$ matrix

Here's how we calculate the inverse of a $2 \times 2$ matrix:

Note: If the determinant of matrix $A$ equals zero, the matrix is non-invertible.

Example

Let's consider the following matrix:

We can calculate the inverse of matrix $A$ using the following steps:

The inverse matrix will be as follows:

The inverse of a $3 \times 3$ matrix

The inverse of a $3 \times 3$ matrix can be calculated using any of these two methods:

• The Gauss Jordan method

• The Laplace expansion

The Gauss Jordan method

The Gauss Jordan procedure is an efficient and quick method to calculate the inverse of a matrix. It starts with two matrices, the invertible matrix $A$and the identity matrix $I$. The identity matrix is transformed into the inverse matrix later:

Let's consider the following matrix:

The inverse of matrix $A$ can be calculated in the following way:

We can convert matrix $A$ into an identity matrix, $I$, using the Gauss Jordan procedure in the following way:

The right side of the matrix represents $A^{-1}$.

The Laplace expansion

The Laplace expansion is also known as the cofactor method used to calculate the determinant of an $n \times n$ matrix and calculate its inverse. $A^{-1}$ is calculated in the following way:

Here the matrix's adjoint is the cofactor matrix's transpose, which is calculated using $C_{ij} = (-1)^{i+j} M_{ij}$.

Example

Let's see an example to clarify the procedure. Let's consider the following matrix:

The determinant of the matrix equals 4.

Note: To calculate the determinant of the matrix in detail, click here.

The matrix of cofactors $C$ can be calculated in the following way:

The adjoint of matrix $A$ is the transpose of the cofactor matrix $C$:

According to the formula, the inverse of matrix $A$ equals the following:

Quiz

Here are a few questions that you can practice: