Inverse Matrix Method

There are two Inverse matrix methods to find matrix inverse:

1. Determinant Method
2. Elementary Transformation Method

Method 1: Determinant Method

The most important method for finding the matrix inverse is using a determinant.

The inverse matrix is also found using the following equation:

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

where,

• adj(A) is the adjoint of a matrix A, and
• det(A) is the determinant of a matrix A.

For finding the adjoint of a matrix A the cofactor matrix of A is required. Then adjoint (A) is the transpose of the Cofactor matrix of A i.e.,

adj (A) = [C_ij]^T

• For the cofactor of a matrix i.e., C_ij, we can use the following formula:

C_ij = (-1)^i+j det (M_ij)

where M_ij refers to the (i, j)^th minor matrix when i^th row and j^th column is removed.

Method 2: Elementary Transformation Method

Follow the steps below to find an Inverse matrix by elementary transformation method.

Step 1 : Write the given matrix as A = IA, where I is the identity matrix of the order same as A.

Step 2 : Use the sequence of either row operations or column operations till the identity matrix is achieved on the LHS also use similar elementary operations on the RHS such that we get I = BA. Thus, the matrix B on RHS is the inverse of matrix A.

Step 3 : Make sure we either use Row Operation or Column Operation while performing elementary operations.

We can easily find the inverse of the 2 × 2 Matrix using the elementary operation. Let’s understand this with the help of an example.

Example: Find the inverse of the 2 × 2, A =  [Tex]\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}[/Tex] using the elementary operation.

Solution:

Given:

A = IA

[Tex]\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}~=~\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}~×~\begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}[/Tex]

Now, R₁ ⇢ R₁/2

[Tex]\begin{bmatrix}1 & 1/2\\ 1 & 2\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\ 0 & 1\end{bmatrix}~×~A [/Tex]

R₂ ⇢ R₂ – R₁

[Tex]\begin{bmatrix}1 & 1/2\\ 0 & 3/2\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\ -1/2 & 1\end{bmatrix}~×~A [/Tex]

R₂ ⇢ R₂ × 2/3

[Tex]\begin{bmatrix}1 & 1/2\\ 0 & 1\end{bmatrix}~=~\begin{bmatrix}1/2 & 0\\-1/3 & 2/3\end{bmatrix}~×~A [/Tex]

R₁ ⇢ R₁ – R₂/2

[Tex]\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}~=~\begin{bmatrix}2/3 & -1/6\\ -1/3 & 2/3\end{bmatrix}~×~A [/Tex]

Thus, the inverse of the matrix A = [Tex] \begin{bmatrix}2 & 1\\ 1 & 2\end{bmatrix}   [/Tex] is

A^-1 = [Tex]\begin{bmatrix}2/3 & -1/6\\ -1/3 & 2/3\end{bmatrix} [/Tex]

The inverse of Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any matrix A, its inverse is denoted as A^-1.

Let’s learn about the Matrix Inverse in detail, including its definition, formula, methods on how to find the inverse of a matrix, and examples.