Inverse of a Matrix

The inverse of a matrix is what results in the identity matrix when multiplied by the original. Note that a matrix can have an inverse only if it has a non-zero determinant, and such matrices are called invertible matrices.

For example:

The inverse of a square matrix A is denoted as A–1.  It is the mathematical opposite of A. If A is a square matrix, then its inverse satisfies A × A–1 = A–1 × A = I

 So, the inverse is:
 

To calculate the inverse of a matrix, we need to learn certain terms related to it:

Minor: To find the minor of an element in a matrix, remove its row and column. Then, calculate the determinant of the remaining smaller matrix.
For example, in the matrix
 

  So, the minor of element a11 is:
 

Cofactor: To find the cofactor of an element, we multiply its minor by (-1)i + j.  Here, i and j denote the element’s row and column positions.  

Determinant: A single value that represents a square matrix is referred to as the determinant. It can be calculated by expanding along any column or row. The determinant is the sum of each element multiplied by its cofactor. Determinant of A = a11 ​× Cofactor of a11 +a12 × Cofactor of a12​ +a13 × Cofactor of a13​. So, det(A)= a11​C11​+a12​C12​+a13​C13​

Singular Matrix: If the determinant of a matrix is zero (i.e., ∣A∣ = 0), the matrix is singular.  An inverse does not exist for a singular matrix.

Non-Singular Matrix: A non-singular matrix always possesses a non-zero determinant and, therefore, such a matrix has an inverse. (i.e.,  ∣A∣ ≠ 0). 

Adjoint of a Matrix: The adjoint of a matrix can be determined by transposing the matrix of its cofactors.
 

We represent the inverse of any square matrix A as A–1. Additionally, the product of a matrix and its inverse always results in the identity matrix I.

The inverse of a square matrix A is given by the formula:

A–1= 1A× adj (A)
Where:
|A| → determinant of matrix A
Adj (A) → adjoint or transpose of cofactors of A

What are the properties of Inverse of a Matrix?

The inverse of the matrix has certain key features, as discussed below:
The inverse of an inverse matrix is the original matrix.
(A–1)-1= A
The inverse of the product of two matrices, A and B, is represented as.
(AB)-1= B–1 A–1
A matrix only possesses an inverse if it is non-singular.
The inverse of the transpose of a matrix is equal to the transpose of its inverse: 
(AT)-1 = (A-1)T

How to Find the Matrix Inverse?

The two common methods to determine the inverse of a matrix are:

Using Matrix Formula:
The inverse of a matrix is the adjoint divided by its determinant.
A–1(inverse of a matrix A) = Adj A/ |A|
 

To find the matrix inverse, we use two Inverse matrix methods, such as:

• Determinant and adjoint Method
• Elementary Transformation Method

Determinant Method:

The inverse of a matrix is calculated using the determinant and adjoint.
The formula for a square matrix A:
A–1 = 1/ det (A) × adj (A)
Where:
A–1 → inverse of matrix A
det (A)→ determinant of A
adj(A) → adjoint of A
If the determinant of A is zero, the matrix has no inverse.

Elementary Transformation Method:

In this method, we find the inverse using a step-by-step process:

Step 1: Express the matrix equation as [A\I]. Here, I is the identity matrix, which is in the same order as the matrix A.

Step 2: To convert matrix A into the identity matrix on the left, use only row operations or only column operations. Apply the same step to the identity matrix on the right side.
The matrix B on the right is the inverse of A once you obtain: I = B × A.

Step 3: Do not combine row and column operations; instead, only use one of them.
Using this method, we can quickly determine the inverse of a 2 × 2 matrix.

2 × 2 matrix Inverse
The inverse of a 2 × 2 matrix can be found using a simple method:
Let’s say:

Since |A| = (ad – bc) 
 
Using the formula:
A–1= (1/ |A|) × adj A

So, 

Hence, the inverse of the 2 × 2 matrix is determined.

Inverse of a 3 × 3 Matrix
In the case of any 3 ×  3 matrix:

Use the inverse matrix formula:
A–1= 1/ | A| × adj (A)

Determinant of Inverse Matrix

The determinant of an inverse matrix is derived by taking the reciprocal of the original matrix’s determinant:
det(A⁻¹) = 1 / det(A)

Proof (in simple steps):

We know: 
det(A × B) = det(A) × det(B)

Also,
A × A⁻¹ = I (by inverse matrix property)

So, 
det(A × A⁻¹) = det(I)

det(A) × det(A⁻¹) = 1 (since det(I) = 1)

det(A⁻¹) = 1 / det(A)
Hence, proved.
 

The inverse of a matrix is not just a mathematical concept; it has numerous practical uses beyond math. Here are a few real-life instances where it can be applied.

• Engineers and scientists use inverse matrices to solve practical problems, such as designing machines or mixing the right proportions of ingredients.

• Businesses use matrices to plan the production and sales of their products.

• This concept is widely used by robots and GPS systems to determine their location or to return if they take a wrong turn.