Let A = [a_{ij}] be a square matrix of order n and let C_{ij} be cofactor of a_{ij} 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 = [C_{ij}]^{T} => (adj A)_{ij} = C_{ji}

**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^{–}^{1}A^{–}^{1}. In general, if A, B, C are invertible matrices then

(ABC…..)^{–}^{1} =…..C^{–}^{1}B^{–}^{1}A^{–}^{1}

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

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