Question

(1 point) Let 3 -4 A = -4 -1 -4 -2 -2 If possible, find an invertible matrix P so that D = P-1 AP is a diagonal matrix. If it is not possible, enter the identity matrix for P and the matrix A for D. You must enter a number in every answer blank for the answer evaluator to work properly. P= II II D= Be sure you can explain why or why Is A diagonalizable over R? diagonalizable not.

Answer 1

Given that A = 3 4 - 4 -4 1-4 -1 1-2-2 charecteristic equation of A is given by 13-2 4 -4 I-4 -1-X - 4 -2 - 2 1-X > (3-x) {-(***) (-2) - 8} -43-4(1-2) -83 -48-2() = (2-x){x?-1-83 -uş q-123-456-2x) = 0 (3+x)(3979) – 16x +48 - 24 + 8x = 0 10 322-27-x² +9x -16% top8 +24+8x20 3x - 27x²+x+2420 ㅋ 3x - x² 3+x-320 x²-3x -x+3=0 = x= 3,-1, I Hence eigen values of a of a are 3,-1,1 an eigen rector Let (2, 4, s )T is corresponding eigen ralue X=3

Then 4 - -4 -4 -4 r - 2 - 2 13 O ч , -чи, 20 7 U = Uz -O -424,-4 4-4 U₂ - 7 +ų + z = 0 + 9 = -25, when ų ? Urs so is an eigen rector of A corresponds x= 3 2 veeter Similarly is an eigen 2 x=1 corresponding and 2 is eigen rector corresponding non Therefore "2 - 2 0 0 2 2 P D = 0 0 3 Since all eigen values of A are dinctinct, So A is diagoni zable over R.