Invertible Matrices: Overview, Questions, Preparation

A square matrix (A) of n x n order is called invertible if and only if its inverse matrix (B) exists such that AB = BA =  $I_n$ . Mathematically inverse of matrix A is denoted as  $A^{-1}$ . The inverse matrix will be of the same order as the original matrix.

These invertible matrices whose inverse exists must be non-singular matrices as their determinant exists. A square matrix which is singular o degenerate is not invertible, since its determinant is zero.

You must understand that inverse of a matrix can only exists if and only if the matrix is non-singular. This means the determinant of the matrix must not be zero.

Simplest Method to check whether matrix is invertible or not, it to find determinant.

The formula to find inverse of a non-singular matrix A is given by, A-1 =  $\frac{1}{\left|A\right|}$ (adj A)

Adj(A) = Adjoint of the matrix

|A| = Determinant of matrix

There are several methods for finding the inverse of a matrix. We must first ensure that the given matrix is non-singular by finding the determinant, which should not be zero. We can use the following methods to find the inverse of the matrix.

Eigen Decomposition Method

Gaussian Elimination Method

Newton’s Method

Cayley-Hamilton Method

NCERT Theorem 1: Inverse of a square matrix, if it exists, is unique.

Proof: f Let A = [  $a_{ij}$ ] be a square matrix of order m x m. If possible, let B and C be two inverses of A. We shall show that B = C.

Since B is the inverse of A

AB = BA = I ... (1)

Since C is also the inverse of A AC = CA = I ... (2)

Thus B = BI = B (AC) = (BA) C = IC = C

NCERT Theorem 2: If A and B are invertible matrices of the same order, then (AB-1) = B^-1.A^-1

Proof: From the definition of the inverse of a matrix, we have

(AB)  ${\left(AB\right)}^{-1})$ = 1

or .A^-1 (AB)  ${\left(AB\right)}^{-1}$ = A-1I (Pre multiplying both sides by A^–1)

or (A^-1A) B  ${\left(AB\right)}^{-1}$ = A-1 (Since A-1I = A^-1)

or IB  ${\left(AB\right)}^{-1}$ = A^-1

or B  ${\left(AB\right)}^{-1}$ = A^-1

or B^-1B  ${\left(AB\right)}^{-1}$ = B-1A^-1

or I  ${\left(AB\right)}^{-1}$ = B-1A^-1

Hence  ${\left(AB\right)}^{-1}$ = B^-1A^-1

Properties of Invertible Matrix

Every invertible matrix must be a square matrix. To have an inverse, a matrix must be non-singular.

An invertible matrix will always have a non-zero determinant. The determinant is a scalar value calculated from the elements of a square matrix.

An invertible matrix has a unique inverse. The inverse of the matrix A is denoted by A⁻¹. This relationship is expressed as A.A⁻¹ = A⁻¹.A = I

The product of two Invertible matrices is also an invertible matrix. If A and B are invertible matrices, then their product (AB), and its inverse is given by (AB)⁻¹ = B⁻¹A⁻¹. 

The inverse of the transpose of an invertible matrix is equal to the transpose of its inverse: (Aᵀ)⁻¹ = (A⁻¹) ᵀ

Invertibility of a matrix only hold true if and only if it is a non-singular (non-zero determinant) matrix.

Determinant of Invertible Matrix using the cross multiplication method discussed in class 11 maths determinant chapter.

It is to note that if B is the inverse of A, then A is also the inverse of B : AB = BA = I then, A^-1 = B  $\iff$ B-¹ = A.

The inverse of a square matrix A exists if and only if A is non – singular matrix.

For every invertible matrix, the determinant of the matrix can never be zero.

Every invertible square matrix can have only one or unique inverse matrix.

The inverse of a product of invertible matrices is the product of their inverses in reverse order: (AB)⁻¹ = B⁻¹A⁻¹.

For any non-singular square matrix A, we have (AT)^-1 = (A^-1) T