What do you wish to calculate :

The determinant of a matrix is not quite as simple. For a n x n matrix the definition of the determinant is as follows :

det(A) = Ê (±)a_1j1 a_2j2. . .a_njn

where Ê is our summation over all permutations j₁ j₂ ... j_n of the set S={1, 2, ..., n }. The sign is + or - according to whether the permutation is even or odd.

In our 3x3 case it is a little easier, and boils down to :

det(A) = aei + cdh + bfg - ceg - bdi - afh

where are matrix first row is a b c , 2nd row d e f, and 3rd row, g h i

The adjoint of A is the matrix whose i, and jth element is the cofactor A_ji of a_ji

The cofactor of an element a_ij = (-1)^{i + j} * det (A'). where A' is the matrix obtained from "omitting" the ith and jth rows.

The inverse of A is the matrix which when multiplied to A returns the identity matrix. The inverse was obtained using the Theorem :

Aadj(A) = det(A)I_n

Which when manipulated gives you : A^-1 = (1 / det(A)) * adj(A)