Question

For the matrix A, find (if possible) a nonsingular matrix P such that p-AP is diagonal. (if not possible, enter IMPOSSIBLE.) 2 - 2 3 A= 0 3-2 0-1 2 PE 11 Verify that p-TAP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP - 11

Answer 1

2 -2 3 A - 2 Charalestic equation (A-131–0 . 2 -> - 2 3-7 -2 2-7 ---3/8=-»1-2) +L•)*s[0]=0 (2->)[6-37-22 +24-2) = 0 (2-[ -57+40 -73+51? -47+272-109+8 20 -23 +70²_147+8=0 solvig et we get, 21=4, 7271,7332 Theis an the eigenvales. Now frid eyen vectors for 1,=4 - 2 2-0 3 o 3-7 6 - 1 2-7

2 - 2 3 - 2 O 6 perfoun R3 B3-R2 - 2 - 2 3 - 2 o hee con coute et in ondrix Jon, -2 - 2 3 2 O ។ -0 ) Let 2=t then -y-22=0 .: -4 = 22 Y = -22 = -2+ and - 2x - 2 + 3 2 = 0 -2x = - 32 +29 5-30 +2 (-2+) -2x =-34-4+ = -70 .:X- - 7 / 2 + / 2 + 1/2 as the ergenvedter qp=4

For d2=1 - 2 3 0 2 - 2 6 - perform R3 = R₂ + R2 2 3 2 o 2 0 he can heat in merix Jomar, - 2 - 2 - 2 13 2 6 Let 2- t 24 +-2 22 = 0 24 = 22= 2xt Ty=t and X - 24 +32 co = 27-32 = 2t-34 1x E-t X - ..Xa veetu & A=1 is the egin veetu

for 7=2 perform 0 Ri - 1 / 1 - 2 3 - 2 o o 1-3/2 oo -12 . o perfon R2 R2 + RI 2 3 perform R,R,-322 -Y2 o -12 0 . and R3 R3-R1 In madrix Joum, 2 3 o Х -Y2 -1/2 - 3/2 2 and R3 R3 - 3R2 Letx=t then, - 2 3 Y=0, 2=0 - -Y2 o X- o o

therefror malorx 'p can be written an Fila - 1 P= - 2 :) 0 Find prl In augmented Join mainox'p can be wretters an -1 1 00 1 2 3 - o O -2 O O 01 perfon R, ZR, - -2/ 29|h OD -2 0 O - 6 1 R2 R2 +2 R 0 1 -2/ h 12ho 37 uh uh

prfon K3 - — — - 1 •17 за ча -17 о 47 о б 17 - 17 • 2/a Рим из чег - 322 - 2, +7 | 1 о 3) 11 4), – 2. -2 -3 2 peform R 27 +7 к. — - 2/7 °/, 1/3 2) 43 7/4 6 2 - 2 р» п ҡ, әк, + 21, 2 1 б 2/3 413 2/3 21, о ч/ 3 7/3 о – 3 ) о 2 - 2 к з R3 -l3 | 213 214 о ul 3 | 4), 1/3 о 32 -і, о

perfon R, > R, م (به 1 ч з 6 - 14 43 1/3 о -/, - о — Эlo ром м K, PR, - e@o -/+ 3 о о V, V, 3, -у, с. р- о - - /s 2 6 12 2/3 3), -/ Now verby ptap 2. – 2 3 7/2 - | 6 - 'lg 73 з ча +12 -2 3 - 2 - – І а. б -ЧІз ч/з 1]. - 1) - p1ap - б 3 2/3 • 2 2 З -) (

.: ptap O 0 O o 2 Her prap is Simler to Ceagonal of & the matrix A. Hence verfied eyenrales

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