Inverse of a Matrix

For a given square matrix A = ǀǀa_ijǀǀⁿ₁ of order n there exists a matrix B = ǀǀb_ijǀǀⁿ₁ of the same order (called inverse matrix) such that AB = E, where E is the unit matrix; then the equation BA = E also holds. The inverse of a matrix A is designated as A^–1. For the existence of the inverse of a matrix A^–1, it is necessary and sufficient that the determinant of the given matrix A be nonzero; that is, the matrix A must be nonsingular. The elements b_ij of the inverse of a matrix are found by the formula b_ij = A_ji/D, where A_ji is the cofactor of the element a_ij of matrix A and D is the determinant of matrix A.