Inverse of a Matrix

•    Inverse of a square matrix is same as the reciprocal of a number. Inverse of square matrix A is denoted as A–1.

•    Product of a square matrix and its inverse is always a identity matrix of the same order that is AA–1 = I.

•    All invertible matrices are square matrices. However, it is not necessary that all square matrices are invertible.

•    Elementary operations on a matrix are as follows:

1. The interchange of any two rows or two columns.
2. The multiplication of the elements of any row or column by a non zero number.
3. The addition of the elements of any row or column to the corresponding elements of any other row or column multiplied by any non-zero number.

•    Square matrix “A” of order n is invertible, if there exists a square matrix “B” of the same order such that   AB = BA = I .In this case, A is said to be invertible and B is called inverse matrix of A and it is denoted by A-1. In other words, B = A-1 and A = B-1

•    Inverse of a square matrix, if it exists, is unique.

•    If A and B are invertible matrices of the same order, then (AB)–1 = B–1 A–1.

•    If A is a matrix such that A-1 exists, then to find A-1 by elementary row operations, write A = IA and apply a sequence of row operations on A = IA  till we get I = BA.The matrix B will be the inverse of A.

•    A square matrix A is said to be singular if |A| =0.