Algebra 2 - Identity & Inverse Matrices



  • A square matrix in which the numerical value of all the diagonal elements is 1 and the remaining elements are 0 (zero), is called an Identity Matrix.
  • A n×n identity matrix is represented as In.
    • An identity matrix of order 2 is written as I2=1001.
    • An identity matrix of the order 3 is written as I3=100010001.
  • Properties of Identity Matrix: 
    • It is always a square matrix.
    • If we multiply any matrix by the identity matrix, the result will be the matrix itself.
    • The product of two inverse matrices yields an identity matrix.



  • The inverse of a matrix A is denoted by A-1.
  • It is defined only for square matrices.
  • If A is a square matrix of order n, and if there exists another square matrix B of the same order n, such that AB=BA=In, then B is called the inverse of A or reciprocal of A. Here, In is an identity matrix of order n.
    • If B is the inverse of A, then A is also the inverse of B.
  • The inverse of a square matrix exists if and only if it is non-singular i.e. A0.

Finding Inverse of a Matrix

  • We have seen that the inverse of a square matrix exists only if the given matrix is non-singular. So, the first step towards finding the inverse of a matrix must be to check whether the given matrix is non-singular or not.
  • There are two methods for finding the inverse of a given non-singular matrix:
    1. by elementary (row or column) operations
    2. by adjoint method

1. Finding the inverse of a matrix by elementary operations

    • If A is a non-singular matrix, then in order to find A-1 using elementary row operations, we first write A=IA and then apply a sequence of row operations on A=IA till we get, I=BA. The matrix B will be the inverse of A.
    • Similarly, if we wish to find A-1 using column operations, then, we write A=IA and apply a sequence of column operations on A=IA till we get, I=BA.
    • Refer to example 2 for a better understanding.


2. Finding the inverse of a matrix by the adjoint method

  • The formula to find the inverse of a non-singular matrix A is given by: A-1=adj AA, where adj A means adjoint of A.
  • Refer to example 3 for a better understanding.

Solved Examples

Example 1: Find the inverse of a matrix A=256152×2 using a suitable method.

Solution: First of all, we must check whether it is non-singular or not. To do so, we will find out A. We know that for a non-singular matrix A0.


Since A=0, the inverse of the given matrix doesn't exist.


Example 2: Find A-1 if it exists for a matrix A=-142210102.

Solution: Let us check whether the given matrix is non-singular or not. For that, we evaluate the value of A.


Since A0, inverse of the given matrix exists. Now, it is up to us to select a convenient method to proceed ahead.

Let us find the inverse by elementary row operations.

First, we write A=IA.

or -142210102=100010001·A

Applying R1R1-R3, we get -240210102=10-1010001·A

Applying R1R1-4R2, we get -1000210102=1-4-1010001·A

Applying R1-110R1, we get 100210102=-11025110010001·A

Applying R2R2-2R1, we get 100010102=-110251101515-15001·A

Applying R3R3-R1, we get 100010002=-110251101515-15110-25910·A

Applying R312R3, we get 100010001=-110251101515-15120-15920·A

Thus, A-1=-110251101515-15120-15920


Example 3: Find A-1, if it exists, given A=451-102536.

Solution: First, let us find A.


Since A0, the inverse of the given matrix exists. Let us find the inverse by the adjoint method.

We know that A-1=adj AA. We have already found that A=53. We need to compute adj A.

Let us find the co-factors.










adj A=A11A12A13A21A22A23A31A32A33=-6-27101619-9-3135

Now, A-1=adj AA=153·-6-27101619-9-3135=-653-2753105316531953-953-3531353553

Cheat Sheet

  • The inverse of a matrix is unique.
  • The inverse of a [2 x 2] matrix: abcd-1=1ad-bcd-b-ca
  • A-1=adj AA
  • AB-1=B-1A-1
  • A-1-1=A
  • AT-1=A-1T

Blunder Areas

  • An identity matrix can never be a rectangular matrix. It is always a square matrix.
  • The inverse of a matrix exists only for non-singular matrices. So, one must verify the same before actually jumping toward finding the inverse of a given matrix.