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Last updated on July 5th, 2025

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Inverse of a Matrix

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The inverse of a matrix A can be represented as A–1. We use a direct formula to calculate the inverse of a 2 × 2 matrix. However, when the order increases, i.e., to 3 or higher, we need to compute the determinant and the adjoint of the matrix. The matrix inversion method is useful for solving linear equations effectively. In this article, we will discuss the inverse of a matrix and its wider applications in real life.

Inverse of a Matrix for Indian Students
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What is the Inverse of the 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:
 

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What is the Formula for Inverse Matrix?

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|
 

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Using Inverse Matrix Methods:

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.
 

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Real-Life Applications of Inverse of a Matrix

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. 
     
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in the Inverse of a Matrix

The inverse of a matrix is an important concept in mathematics. However, students often make mistakes while calculating it. Here are a few common mistakes and tips to avoid them.

Mistake 1

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Attempting to find the inverse of a non-inversible matrix
 

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Not all matrices have an inverse. Students may forget this and attempt to find the inverse of a matrix with a zero determinant. This results in undefined or incorrect values.
For example, the matrix A 

Calculate the determinant det(A) = (2)(2) - (4)(1) = 4 - 4 = 0.
Students may directly start calculating the inverse without realizing that it is a singular matrix.
Always calculate the determinant and make sure that the matrix is square before trying to find its inverse.
 

Mistake 2

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 Incorrectly calculating the determinant
 

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Computing the determinant is the first step for finding the inverse of a matrix. Miscalculating the determinant can affect the inverse.
For example, for a matrix B,

The determinant is det(B) = (3)(4) − (1)(2) = 12 − 2 = 10
If the value is calculated incorrectly, for instance det(B) = (3)(2) − (1)(4) = 6 − 4 = 2, this wrong value will impact the result of the inverse. So, always make sure to calculate the determinant correctly and recheck for its value to be sure.
 

Mistake 3

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 Errors in forming the Adjugate matrix
 

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The adjugate matrix is formed by calculating the cofactors of each element of a matrix and then computing the transpose of the cofactor matrix. Students may get confused and make errors while following this process. 
For example, let us take a matrix c

It’s cofactor matrix is 

The adjugate matrix is the transpose of this cofactor matrix

This is the correctly computed adjugate matrix. However, the most common error in this process is the sign error.
Students may write the adjugate matrix as

So, always make sure to verify the transpose and follow the checkerboard pattern for assigning signs.
 

Mistake 4

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Misapplying Gaussian elimination or row reduction
 

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Errors in row operations like row reduction in Gaussian elimination result in an incorrect identity matrix, resulting in a wrong inverse.
For a given matrix D,

We augment it with the identity matrix and apply row operations:

If we accidentally subtract the second row from the first instead of scaling properly, 
R1​ → R1 ​− R2 ​⇒ Wrong transformation. This leads to an incorrect inverse matrix. So, systematically perform row operations and recheck each step.
 

Mistake 5

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Not verifying the inverse
 

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Once you calculate the inverse, it is important to verify it to ensure that the result satisfies the condition A  A-1 = I


 

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FAQs on Inverse of a Matrix

1.What do you mean by the inverse of a matrix?

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2.Does every matrix have an inverse?

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3.What does it mean when the determinant is zero?

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4.How is a matrix related to its inverse?

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5.Give the formula for the inverse matrix.

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6.How does learning Algebra help students in India make better decisions in daily life?

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7.How can cultural or local activities in India support learning Algebra topics such as Inverse of a Matrix ?

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8.How do technology and digital tools in India support learning Algebra and Inverse of a Matrix ?

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9.Does learning Algebra support future career opportunities for students in India?

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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