The inverse of a square matrix A is a second matrix such that AA-1 = A-1A = I, I being the identity matrix. There are many ways to compute the inverse, the most common being multiplying the reciprocal of the determinant of A by its adjoint (or adjugate, the transpose of the cofactor matrix). For example,
This is indeed the inverse of A, as
A matrix is invertable if and only if the determinant is not equal to zero.
The inverse of a matrix is normally defined for square matricies. For non-square matrix, a corresponding pseudoinvere matrix can be constructed to produce an identity matrix.