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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 = B–1A–1. In general, if A, B, C are invertible matrices then
(ii). The inverse of the inverse of the matrix is the original matrix itself, i.e. (A–1)–1 = A.
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