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