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

Also Read : How to find Adjoint of the Matrix (2×2 & 3×3)

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. \(|A^{-1}|\) = \(1\over |A|\).

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