Question

Diagonzalize the matrix A.

if possible. That is, find an invertible matrix P and 1 3 3 Diagonalize the matrix A= - 3 - 5 -3 3 3 a diagonal matrix D such that A = PDP-1. 1

Answer 1

Given matrix: ri ww -5 -3 3 1 The characteristic polynomial: 3 -3 3 -5- 3 1-1 2 + 4 -3-322 = -(4-1 69-74*+4) = -(4-1) (0+2) (174) Hence, a;= 1, 22=-2 as (x-1) = 0 4 (x+2) = 0 For a = 1 M 3 A- x, I -3 -6 - 3 3 O set to systems of ع لر have A2 = iv > (4 – 17) B=0 homogeneous we sore it by Gaussian Blimation! linear equations o 3 -6 -3 3 - 6 - 3 ? 3 R2 Ry 3 2 3 Ri/(-3) ->RI O 3 O o

For 22 = -2 A dz 1 -2 -3 I det av = xv So we have ous system of >> (4-73) 050 homogeneon equations we solve it by Gaussian Elimination: lincar 3 mm 3 -3 -3 3.3 3 R1/3 - R -3 R2-(-3) R, R2 3 - R3-3R, R3 10 M tuz tn 3 - 0 agnez General solution: 4 nz M M of M 1 O gros are rectors Hence, three 1 ₂ - ; V3 wa (37 و Na 1-1 1