Question

11 (10) [10pts) Let A be the 2 x 2 matrix A-L 5 A = 2 4 The characteristic polynomial of A is (3- A) (1 - ). Compute an invertible matrix P and diagonal matrix D such that A= PDP-

Answer 1

Let A be the 2 times squareroot 2 matrix A = [5 -4 2 -1] The characteristic polynomial of A is (3 - lambda) (1 - lambda) compute an invertible matrix P and diagonal matrix D such that A = PDP^-1 solution Let A = [5 -4 2 -1] The characteristic polynomial of A is (3 - lambda) (1 - lambda) = 0 root of characteristic poly are 3 & 1 Therefore eigen value of A is 1 and 3 which are different Now we find eigen vector corresponding eigen values For lambda = 1 (A - lambda I) v = 0 [5 - lambda -4 2 -1 - lambda] [x y] = [0 0] lambda = 1 = [4 -4 2 -2] [x y] = [0 0] \r\n R_2 rightarrow R_2 + R_1 [4 0 2 0] [x y] = [0 0] Therefore y is free variable put y = k_1 k_1 elementof 1R