Inverting 3x3 Matrices. The signs used are in a chess board pattern: . Thus A2 the cofactor of a2 is. . . Etc.. The determinant can now be defined in terms of the cofactorsas: det = a1A1 b1B1 c1C1or det = a1A1 a2A2 a3A3det = a2A2 b2B2 c2C2or det = b1B1 b2B2 b3B3det = a3A3 b3B3 c3C3or det = c1C1 c2C2 c3C3.