Question

Problem #9: Consider the following matrix. A = i 1 07 1 1 0 1 1 0 Find a matrix P that diagonalizes A. [1 1 0 1 o 1 11 [1 0 17 o 1 -1] -1 -1 11 ſi 0 17 (A) 0 1 -1 (B)0 1 -1 (C) 0 0 1 (D) 1 1 0 (E) 0 1 1 (F) 0 1 1 LO 11] [110] LO 1 1] LO 11] [101] [O0 1] To 0 17 (G) 0 1 1 | 1 0 1

Answer 1

Consider matrix A= 1 a He find matrix p that dragon alize Where P is matrix of eigenvectors First finding eigen values. consider PA-TI1=0 I-d 01 FI-> ozo (I--) [ -=(1-0)] - 1[-a-c] =0 -A(1-)(1-2) +@=0 -a [(1-x)(1-2) € -1) = 0 il 1-2x+d2-, 3-0 al a2-22] co a. ² [a-z] to > -0,0,2 a = pz 20 & 23=2 are ergen values kinding eigen vectors ... Nou considers (4-AI)X =0 i olx LI O Augmente nga 1 1 0

form Reducing to reduced row echilon R2R [110107 R3- R 00010 3) 24-4.82 = 0 xg is free cs (801) as as free variable - (san) 39=+t 24 Itt 17 lol -- E t- Islo SJ LOL Thus Thus via [*] ... =(8) V -1 , V O - ® for daz Consider Arad) x 20 -1 1 0 71 647-18 1 -2 ) Augmented matris is - 10 Reducing -IR to reduced row 1-1 010) echilon forros Ra-R to 2 -20 -OO-SO - Tod To- le 1 - 0107

[ 1 -1 0,07 I Lolo 0 TL c. x,-X₂20, Xe ~X320 ser to as free rasiable say x₂=t <. x, ct, X2=t rx, 7 st1 l 1 X2 = t lat - [X3] It + V3+ [: :: Par ve Ve Vj] on palve vz vs] P = Foi 011 0 ; - Li i oj is required matrix.