Question

ark Sets iagonalization oblem 8 5.4 Diagonalization: Problem 8 ettings Previous Problem List Next Results for this submission Problems Entered Answer Preview Result 9 10 20 9 -1827 -30 10 20 -27-30 10 23 18 0 incorrect 0 10 23 em 1 blem 2 slem 3 blem 4 blem 5 blem 6 oblem 7 oblem 8. The answer above is NOT correct (1 point) CHALLENGE PROBLEM Find a 3 x 3 matrix A whose 3 eigenspace is V = {(1,2,. :) In R-3x - 5y - 10:0} and whose1-eigenspace is W Span -{3} 9 10 20 18 -27 -30

Answer 1

- Find a 3x3 materin A whose 3-eigenspace is v= {(x, y, z) ER3 / -3x - 5y - loz=0} and whose -l eigenspace is wo span -3 of 30 za 10 fy -0,o, -30 10 nty Hence Basis a 3-eigenshace -30-54-107 -lo z = 3x+ 5y 3n - 1 y (als Z) = (ny, -30 (*:0, -3x) + (0.yout 11,0,-3)* y losis va shan (1,01-23 (0, shem {110, 0, -3), (0,2,-13 Geometric multiplicity of eigenvalue 3-2. So, for diagonalization algebraic multiplicity also must be 2. Hence there must be two eigenvaluer 3,3 and eigenvector (10,0,-3), (0, 2, -1). Third eigenvalue is -l and eigenvectors (1,-3,1). Hence Corres responding

Now for some A invertible matrin e and diagonal mataina pol AP- رم 0 AP= PO X Here por lo 0 3 12 2 -3 o 3 - 3 1 o eigenvalus Columns are genvectors

Inverse of p 10 : 0 2 -3 ; O -3 -1 o o R3 3 R1 tlo R₃ lo ОО o 2 -3 ܙ ܢ wo -10 13 10 R 5 R2 + R3 10 o o 2 -3 © -2 3 5 10 Ri 2 Rot R3 R2 2 R2 - 3R3 o 5 5 10 Ч -9 -13 -30 . o -2 3 5 10 Ry - RI R2 - R2 R3 R3 20 ។ -2 o 0 Yy -१५ -3/2 ''y / -1374 -1/2 -512 s Heru pts - t st 1 2 늘 -13ly alsh - 5/2 - 5 aq -13 -30 -112 -6 -10 -6 -२०

P-A POD As Pop-l 프 10 ㅣ 2 10:10: 29 -13 -30 6 - 10 - 20 30 - 2 (4) 5 3 9 -13 -30 -1 → 3 ~ ) 6 - to -20 () 3046 3otlo 60t20 -54-18 - 78 -30 -1go - 60 927 6 9 139 tlo - 1% - Io+26 E 36 uo go 72 108 ~ 20o २५ uO ~88 10 २७ 10 18 ~27 -60 10 -22 Hence Required matain matain to 여 (o 20 - 18 -27 60 10 0 -22