Let A = [aij] be a square matrix of order n and let Cij be cofactor of a­ij in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A.

Thus, adjA = [Cij]T  => (adj A)ij = Cji

Properties of Inverse of a Matrix:

(i).       (Reversal Law) If A and B are invertible matrices of the same order, then AB is invertible and (AB)1 = B1A1. In general, if A, B, C are invertible matrices then

(ABC…..)1 =…..C1B1A1

(ii).       The inverse of the inverse of the matrix is the original matrix itself, i.e. (A–1)–1 = A.