Inverse Matrix

Before calculating the inverse of a matrix let us understand what a matrix is? A matrix is a definite collection of objects arranged in rows and columns These objects are called elements of the matrix. The order of a matrix is written as number rows by number of columns. For example, 2 × 2, 2 × 3, 3 × 2, 3 × 3, 4 × 4 and so on. We can find the matrix inverse only for square matrices, whose number of rows and columns are equal such as 2 × 2, 3 × 3, etc. In this article, you will learn what a matrix inverse is, how to find the inverse of a matrix using different methods, properties and examples in detail.

Table of Contents:

Matrix Inverse

If A is a non-singular square matrix, there is an existence of n x n matrix A-1, which is called the inverse matrix of A such that it satisfies the property:

AA-1 = A-1A = I, where I is  the Identity matrix

The identity matrix for the 2 x 2 matrix is given by

identity matrix 2 x 2

It is noted that in order to find the inverse matrix, the square matrix should be non-singular whose determinant value does not equals to zero.

Let us take the square matrix A

identity matrix 2 x 2

Where a, b, c, and d represents the number.

The determinant of the matrix A is written as ad-bc, where the value of determinant should not equal to zero for the existence of inverse. The inverse matrix can be found for 2× 2, 3× 3, …n × n matrices. Finding the inverse of a 3×3 matrix is a bit more difficult than finding the inverses of a 2 ×2 matrix.

Inverse Matrix Method

The inverse of a matrix  can be found using the three different methods. However, any of these three methods will produce the same result.

Method 1:


Similarly, we can find the inverse of a 3×3 matrix by finding the determinant value of the given matrix.

Check out: Inverse matrix calculator

Method 2:

One of the most important methods of finding the matrix inverse involves finding the minors and cofactors of elements of the given matrix. Observe the below steps to understand this method clearly.

  • The inverse matrix is also found using the following equation:

A-1= adj(A)/det(A),

          where adj(A) refers to the adjoint of a matrix A, det(A) refers to the determinant of a matrix A.

  • The adjoint of a matrix A or adj(A) can be found using the following method.

          In order to find the adjoint of a matrix A first, find the cofactor matrix of a given matrix and then   

          take the transpose of a cofactor matrix.

  • The cofactor of a matrix can be obtained as

Cij = (-1)ij det (Mij)

Here, Mij refers to the (i,j)th minor matrix after removing the ith row and the jth column. You can also say that the transpose of a cofactor matrix is also called the adjoint of a matrix A.

Similarly, we can also find the inverse of a 3 x 3 matrix. Here also the first step would be to find the determinant, followed by the next step – Transpose.

Method 3:

Finding an Inverse Matrix by Elementary Transformation

Let us consider three matrices X, A and B such that X = AB. To determine the inverse of a matrix using elementary transformation, we convert the given matrix into an identity matrix. Learn more about  how to do elementary transformations of matrices here.

If the inverse of matrix A, A-1 exists then to determine A-1 using elementary row operations

  1. Write A = IA, where I is the identity matrix of the same order as A.
  2. Apply a sequence of row operations till we get an identity matrix on the LHS and use the same elementary operations on the RHS to get I = BA. The matrix B on the RHS is the inverse of matrix A.
  3. To find the inverse of A using column operations, write A = IA and apply column operations sequentially till I = AB is obtained, where B is the inverse matrix of A.

Inverse of a Matrix Formula

Let \(A=\begin{bmatrix} a &b \\ c & d \end{bmatrix}\) be the 2 x 2 matrix. The inverse matrix of A is given by the formula,

\(A^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d &-b \\ -c & a \end{bmatrix}\)

Let \(A=\begin{bmatrix} a_{11} &a_{12} & a_{13}\\ a_{21} &a_{22} &a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\) be the 3 x 3 matrix. The inverse matrix is:

Inverse of 3x3 matrix formula

Inverse Matrix 2 x 2 Example

To understand this concept better let us take a look at the following example.

Example: Find the inverse of matrix A given below:


Inverse Matrix 3 x 3 Example


Inverse matrix 3x3 question


Determinant of the given matrix is

Let us find the minors of the given matrix as given below:

Inverse matrix 3x3 question sol2

Now, find the adjoint of a matrix by taking the transpose of cofactors of the given matrix.

Inverse matrix 3x3 question sol3


A-1 = (1/|A|) Adj A

Hence, the inverse of the given matrix is:

Inverse matrix 3x3 question sol2


A few important properties of the inverse matrix are listed below.

  • If A is nonsingular, then (A-1)-1  =  A
  • If A and B are nonsingular matrices, then AB is nonsingular. Thus, (AB)-1  =  B-1A-1
  • If A is nonsingular then (AT)-1  =  (A-1)T
  • If A is any matrix and A-1 is its inverse, then AA-1 = A-1A = In, where n is the order of matrices

Practice Problems

  1. Find the inverse of a matrix \(\begin{bmatrix} 1 & 2 &3 \\ 3 & -2 &1 \\ 4 & 1 & 1 \end{bmatrix}\).
  2. Obtain the inverse of the matrix A = \(\begin{bmatrix} 0 & 1 &2 \\ 1 & 2 &3 \\ 3 & 1 & 1 \end{bmatrix}\) using elementary operations.
  3. What is the inverse of a matrix \(\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\)?

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