Inverse of a Matrix

A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = In = BA.

In such a case, we say that the inverse of A is B and we write, A−1 = B

The inverse of A is given by A−1 = adj A/|A|

The necessary and sufficient condition for the existence of the inverse of a square matrix A is that |A| ≠ 0

Properties of Inverse of a Matrix:

(i) Every invertible matrix possesses a unique inverse.

(ii) (Reversal Law) If A and B are invertible matrices of the same order, then AB is invertible and (AB)−1= B−1A−1

In general , if A , B , C, …. are invertible matrices then (ABC…..)−1 = …..C−1B−1A−1

(iii) If A is an invertible square matrix, then AT is also invertible and (AT)-1 = (A-1)T

(iv) If A is a non-singular square matrix of order n. Then |adjA| = |A|n−1

(v) If A and B are non-singular square matrices of the same order, then adj (AB) = (adj B) (adj A).

(vi) If A is an invertible square matrix, then adj(AT) = (adj A)T

(vii) If A is a non-singular square matrix, then adj(adjA) = |A|n−2 A.

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