Formula for Inverse of a Matrix – Properties, Example

Here you will learn formula for inverse of a matrix and properties of inverse of matrix with example.

Let’s begin –

Formula for Inverse of a Matrix

A square matrix A said to be invertible if and only if it is non-singular (i.e. |A| $$\ne$$ 0) and there exists a matrix B such that, AB = I = BA.

B is called the inverse (reciprocal) of A and is denoted by $$A^{-1}$$. Thus

$$A^{-1}$$ = B $$\iff$$ AB = I = BA

We have, A.(adj A) = | A | $$I_n$$

$$A^{-1}$$.A(adj A) = $$A^{-1}$$ $$I_n$$ | A |

$$I_n$$ (adj A) = $$A^{-1}$$ | A | $$I_n$$

$$\therefore$$ $$A^{-1}$$ = $$(adj A)\over | A |$$

Inverse of matrix A is $$A^{-1}$$ = $$(adj A)\over | A |$$

Note : The necessary and sufficient condition for a square matrix A to be invertible is that | A | $$\ne$$ 0

Example : find the inverse of the matrix $$\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$$

Solution : Let A = $$\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$$. Then,

| A | = $$\begin{vmatrix} 2 & -1 \\ 3 & 4 \end{vmatrix}$$ = 8 + 3 = 11 $$\ne$$ 0

So, A is a non-singular matrix ( i.e. | A | $$\ne$$ 0 ) and therefore it is invertible. Let $$C_{ij}$$ be cofactor of $$a_{ij}$$ in A.Then the cofactors of elements of A are given by

$$C_{11}$$ = 4, $$C_{12}$$ = -3, $$C_{21}$$ = -(-1) = 1 and $$C_{22}$$ = 2.

$$\therefore$$ adj A = $${\begin{bmatrix} 4 & -3 \\ 1 & 2 \end{bmatrix}}^T$$ = $$\begin{bmatrix} 4 & 1 \\ -3 & 2 \end{bmatrix}$$

Hence, $$A^{-1}$$ = $$1\over | A |$$ adj A = $$1\over 11$$ $$\begin{bmatrix} 4 & 1 \\ -3 & 2 \end{bmatrix}$$

Properties of Inverse

(1) (Cancellation Law) Let A, B, C be square matrices of the same order n. If A is a non singular matrix, then

(i) AB = AC $$\implies$$ B = C

(ii) BA = CA $$\implies$$ B = C

(2) (Reversal Law) If A and B are invertible matrices of the same order, then AB is invertible and $$(AB)^{-1}$$ = $$B^{-1}$$$$A^{-1}$$.

(3) If A is invertible square matrix, then $$A^T$$ is also invertible and $$(A^T)^{-1}$$ = $$(A^{-1})^T$$.

(4) The invertible of an invertible symmetric matrix is a symmetric matrix.

(5) Let A be a non singular square matrix of order n. Then, | adj A | = $$| A |^{n-1}$$.

(6) If A and B are non singular square matrices of the same order, then adj AB = (adj A)(adj B)

(7) If A is an invertible square matrix, then $$adj A^T$$ = $$(adj A)^T$$.

(8) The adjoint of a symmetric matrix is also a symmetric matrix.

(9) If A is a non singular square matrix, then adj(adj A) = $$| A |^{n-2}$$ A.

Note :  If A is a non singular matrix of order n, then |adj(adj A)| = $$|A|^{(n-1)^2}$$.

(10) If the product of non-null square matrices is a null matrix, then both of them must be a singular matrices.

(11) If A is a non singular matrix, then $$|A^{-1}|$$ = $$|A|^{n-1}$$ i.e. $$|A^{-1}|$$ = $$1\over |A|$$.