Adj(M) = Adjoint matrix of square matrix M:

The short definition is the adjoint matrix A of square n by n matrix M is the transpose of the cofactor matrix of M. M * A = A * M = Det(M) * MatId(n). M may be singular.

Let Mhj denote the (n − 1) by (n − 1) submatrix of M obtained by deleting row h and column j, Then the scalar cij = (−1)^(h+j)*Det(Mhj) is called the cofactor of Mhj in M. The n by n matrix Tran(chj) is called the adjoint of M and denoted Adj(M).

For example if M = {1, 2, 3; 0, 4, 5; 1, 0, 6},

M11 = +Det({4, 5; 0, 6}) = 24,

M12 = −Det({0, 5; 1, 6}) = 5,

M13 = +Det({0, 4; 1, 0}) = −4,

M21 = −Det({2, 3; 0, 6}) = −12,

M22 = +Det({1, 3; 1, 6}) = 3,

M23 = −Det({1, 2; 1, 0}) = 2,

M31 = +Det({2, 3; 4, 5}) = −2,

M32 = −Det({1, 3; 0, 5}) = −5,

M33 = +Det({1, 2; 0, 4}) = 4,

So the cofactor matrix is {24, 5, −4; −12, 3, 2; −2, −5, 4}. The adjoint of M is the transpose of the cofactor matrix

A = Adj(M) = {24, −12, −2; 5, 3, −5; −4, 2, 4}

M * A = A * M = {22, 0, 0; 0, 22, 0; 0, 0, 22}

Det(M) = 22.

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