Question

Find the characteristic equation of A, the eigenvalues of A, and a basis for the eigenspace corresponding to each eigenvalue.

A = Find the characteristic equation of A, the eigenvalues of A, and a basis for the eigenspace corresponding to each eigenvalue. -7 16 0 1 1 005 (a) the characteristic equation of A 2+7 2–1 2–5 = 0 (1 - 5)(1 - 1)(x + 7) = 0 (b) the eigenvalues of A (Enter your answers from smallest to largest.) (11, 12, 13) = -7.1,5 -7, 1,5 (c) a basis for the eigenspace corresponding to each eigenvalue basis for the eigenspace of 11 basis for the eigenspace of 12 basis for the eigenspace of 13 =

Answer 1

2. Answer :) Ginen matrix 1-7 1 6 A= how it's characteristic equation is given by (A-1I 1 1-7X 6 1- ) 0 5-7 o. it's upper trairgular. = (-7-2) (1-) (5-1)=0 C+7 (-1 (1-5)=0 These (x²+61-7)(+-5)=0 > 2 +6 1²-71-582-30X +35=0 -> which is known 73 + x² -37d +35=0 as chonacteristic

equation of A now © from 4 +7 (6-1) (-5) = 0 teet +=-7 or d=1 or d=5 so eigen en values of A are 4 =-7 d = 1,4 1₂ = 5 Say o c) now Gur aim to find out basis of eigen space corresponding to each eigen values =-7 Recallos Eigen epace (E) corresponding to the eigen value t o the

hall space of (A-XI) tie E = NCA-BI) x HA DI)x=0 how For 1,=-7 space is E say eigen E, = N CAT7I) i la 170he=0 og how A+71x where let x= (x, ug 1

we reduced matrix into eachler 1-747) 1-67) 57 be © 6 8 + 12 now we whing row R2 OR, - R2 6 O 47 12 o Rz 12 R₂-47Rz 6 LO 47 LO FO. 6 this the solution y of (A +75k 4713 = of My = 0

and xt 6kg = 0 8. x 1 ) = y X₂ - Dc G this the hall space N (A +71) consists of vectors O for an any scolar E N (A+70) = 8b9y span. o one Spanning het has only non-zero. rector & hence it is lina independent as well. parts for his filhos

Ę say Similarly, For dail, Ez = N (A-1) = x[ (–1)X =o) thing 1-7-1 | 6 1-11 5-| X X2 where X px O -8 6 -- xi X2 =lo O I 0 4 44 =0 fl.xg e 480 =) 113=0 o and -84, +42 + 6x2 = - - 8%,+ xy = 0 8ty 0 X₂ = 0 X2 = X X= x2 8x *18 O for any Tapy scalana,

Ę = N(A-1) = span NCA DESDE 8. clealy linesly independent Hence ol eigen and basis space is how Foz dz = =5. AN :-) say Eigenspace E * (A-SD) Xcof NCA-SIS now 1-7-5 55 - 12 6 = 442 % -12x, + X₂ +6x₂=0 4 -4X₂ +X₂ put x 4x2 then 12x = x₂ +68₂ x2 + 6x 4x₂=251

122 =2542 =) n = 82 25x2 12 252 ni ₂ 12 м; I Ik 125 for any scalar T2 0 M2 to N(A-SI) = Span Li & hence basis of Ez is C Ansit 25