adjugate matrix

the product of a matrix with its adjugate gives a diagonal matrix whose diagonal entries are the determinant of the original matrix. consequently, dividing this product matrix by the determinant produces the identity matrix of that size, which is just a diagonal matrix with all the diagonal entries equal to one. The multiplicative inverse of a square matrix can therefore be found by dividing its adjugate by its determinant, since multiplying this by the original matrix gives the identity.