Question

6. Suppose that A is a 3 3 diagonalizable matrix, so there is an invertible 3 x 3 matrix S and scalars a, b, c so that Let C1,C2, č3 be the columns of S. Use the equation S-1S - I3 to computeSč1, S-C2, S-1c3 Show that B[c1,c2,cs is an eigenbasis for A I3 to computeS-1a

Answer 1

Suppose that A is da 3 times 3 matrix, So there is an invertible 3 times 3 matrix S and Scales a,b,c A = [a 0 0 0 b 0 0 0 c]S^- 1 *c_1,c_2,c_3 are columns of S s^- 1 s =I_3 S^- 1(c_1,c_2,c-3) = (1 0 0 0 1 0 0 0 1) [s^- 1c_1, S^- 1c_2 s^- 1c_3] = [1 0 0 0 1 0 0 0 1] Implies s^- 1c_1 = [1 0 0] s^- 1c_2 = [0 1 0] (2) S^- 1 c_3 = [0 0 1] #{c_1, c_2,c_3} rightarrow c_1,c_2,c_3 are linearly independent. because S is invertible matrix of column on every invertible matrix is linear independent. Ac_1 = s (a 0 0 0 b 0 0 0 c) s^- 1c_1Ac_1 = [c_1,c_2,c_3][a 0 0] Ac_1 = ac_1 + 0c_2 + 0c_3 Ac_1 = ac_1