(a) Find basic solutions of homogeneous system with coefficient matrix A, and ex- press every solution as a linear combination of the basic solutions, where A (b) Evaluate the determinant 16x2x + 3x - 17x3 = 0 - 1 -23 + 3x14x4 +22=0 - do o 1o - A= -3 0 6 -9 9 log: 512 log, 8 CITA 2 2 2-5 -1 0 7 (d) Find all solutions of homogeneous system + -3x + x3 = 0 B = 1 2 (c) Find a cubic polynomial y = ar + br + cx+d which passes through the point (1,3) with slope -5 and through the point (3,-7) with slope -1. 6-8 log, 3 log, 9 1 2 0 1 -1 2 1 3 1 0 -1 0-2 0 1 00 028 (7 marks) (6 marks) (e) Find a basis for ker(T) as a subspace of R5. where T: R5 R defined by T(T) = A7, where 6 -2 2 -2 3 -1 (8 marks) (8 marks) (f) Find the characteristic polynomial, characteristic equation and the characteristic roots for the following matrix: (7 marks) (6 marks)