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:
 The interchange of any two rows or two columns.
 The multiplication of the elements of any row or column by a non zero number.
 The addition of the elements of any row or column to the corresponding elements of any other row or column multiplied by any nonzero 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 A1. In other words, B = A1 and A = B1
• 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 A1 exists, then to find A1 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.
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