Question

5. (a) (10 pts) Find the eigenvalues of A= 6 0 -5 2 12 0 0 2 (b) (6 pts) Find a basis for any one of eigenspaces of A (you may use any eigenvalue you have found in (a)).

Answer 1

А % 5. a) valies To find the eigen 0 Az 6 -5 C 2 o -12 2 (A - A1)* = 0 IA-X110 → gines Characterisha equation. 6- - -5 2-1 o -12 2- -5 6-1 2- 0 -12 2 0 2 - -5 2-) - 1 -12 อ 0- (6-4) (2-1). 1 (12(2--))-0 66-^)(+-44 +12)- 24 +124.0 X-10x² + 16x so x(x² - 10x+16)=0 -) diso da = 10 + V100- by 2,8 = 3 2 die 0, des 2 d3-8 These are the required eigen values А q

value 2 to find b ) the eigen lets consider A the basis for eigen space o to know, E We the corresponding nullspace q is 2 eigen vatie the matrix A - 21 i-e, E N( A - 2I) - 2 o 2 A - 21 - [s 6 -5 2 3)-[ 2 -12 4 -S -12 the mabia A-2I by we reduce element any row operations as A- 21 - 4 -5 -12

R3 4 12 R 5 :) -ک e Let (A-21 ) ' O - noorum the eigen vector Ex () IC 4 O -5 [:] 길 o i oo Ex 47 - 23 o -57, = 0 (- 21,=0, 13=0 and 2 is free . basis Ez is o9 [],+0]