Question

Next Problem (1 point) Suppose 7 A 8 -5 Find an invertible matrix P and a diagonal matrix D so that A = PDP-1. Use your answer to find an expression for AⓇ in terms of P, a power of D, and P-1 in that order. -] 1/2 1 -1 0 -2 2 A6 1 1 0 3 2 -1 Note: In order to get credit for this problem all answers must be correct.

Answer 1

Solution A 8 To find invertible matrixp and dicegonal matrix o let's find eigenvalues and eigenvectors of A. Solve det (A-AI) = 0 + 1 = 0+1 = 0721 = 3, 72 = -1 7-7 8 -5-a1 4 - 4 For i = 3, Solve (A-7, I) U = 0 of N:1) 8 = U, =1, 42=1 So, u= -- for 2 = -1, solve (A-712 I) U=0 aske in-1010 4= 1, U2=2 So, le- 3 P = {4, 0] =) PE 4 2 0 Now A = PDP + ² = (PDP') (PDP) = PDP'p DP = Po? ?' = AG - Pop 729 2 So,