Inverse Matrix

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

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

Let us take the square matrix A

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

The determinant of the matrix A is written as ad-bc, where the value is not equal to zero. 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 difficult than finding the inverses of a 2 ×2 matrices.

Inverse Matrix Method

The inverse of a matrix  can be found using the following methods:

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:

• 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

C_ij = (-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 as 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.

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 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 Matrix Example

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

Example: Find the inverse of matrix A given below:

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