Let A be a square matrix of order n. Then a matrix B, if it exists, such that AB = BA = In is called inverse of the matrix A. In this case, A is an invertible matrix.
If a matrix A possesses an inverse, then it must be unique.
If A is a square matrix of order n, then
A(adj A) = (adj A)A = |A| In
Assume that A is non-singular, then |A| ≠ 0.
If A is a non-singular matrix, there exists an inverse which is given by
1. Reversal Law for Inverses
If A, B are any two non-singular matrices of the same order, then AB is also non-singular and
(AB)−1 = B−1A−1
The inverse of a product is the product of the inverses taken in the reverse order.
2. Reversal Law for Transposes
If A and Bare matrices conformable to multiplication, then
(AB)T = BTAT
The transpose of the product is the product of the transposes taken in the reverse order.
3. For any non-singular matrix A,
(AT)−1 = (A−1)T
For square matrix, find |A|.
If |A| ≠ 0, then A is a non-singular matrix. Hence, it is invertible.