Question

0 0 -2 -1 1 6 4. Consider the matrix D= Show that a=1 and 2=2 are eigenvalues of D, and 2 4 -2 0 -2 1 find their associated eigenspaces.

Answer 1

Solution: Giwn that С -2 -1 DE 6 S 2. 0 4 1 0-2 1 Connicles a matsin. A of cseler nxn if x is an eigenvalue of A Then del (0-dI)=0 when. x=1 (D-C1]) = (0-3) = 165 0 1 0 2 0 4 1 - 2 0-2 l - 6. 5 2 3 - 2 ara we know, by uning Row oposation determinant will not change 2-0,48 R2 + R₂ +2 R, Ryan Rq + 2 R, 4 T &ī i

- 2 Re the 24 R3 O 0 O al -6 -2 Row i Second Row in completedy completedy Zero del (o-slao Hence, Son a=1 an eigenvalue. Now, to get eigen nace corresponding to tal; we need to solwe (0-11) VS0 (Q-I) U=0 ana 91 - 65 2. 12 12 3 - 13 Now, foom (x1 we have Elementary А OS Row operation Recluced form of T -9 - -6 -2 PA RA - 683 160 GI -9 O 9

T - - 22 - MA=0 fil - 13 - na - 4 MA=0 (*) masol - ₂ = 0 20 -H, -0-0 =0 1,20 aand ng is a free variable. Fiepriau Ex=1 M2 GIR Me (IR Now, when A=2 det (1-21) D-29 2 0 - 2 - 1 - - -1 6 5 2 0 4 1 -2 0 g Oolo 0001 2

Now. by Elementary Row operations we have R, 48, 49 lo १ Rza Rg 2R 6 5 R&R + 2 Rq 2. -lo o 10 9 Ry - - RA+R3 lo q - 6. 2 -10 na R 348378 10 9 6 5 e O Riera - 6 5 82 10 9 konce, (det (D-21) = 0 Sun 1=2 il an eigenvalue. Now, get Eigenpal, or equivalenty we need to golue (1-25)v=0 Buzo V 65 M o nq ng 14

Fm О. чеn 10а 15, X - Mg + 6H₂ +544 = 0 fil -2 ngt long +994 = 0 W3 and we are free variables let 14 = 2 =) Hy het61 +5 t = 0 fi) -2xy +10)+96=0 fii) from liil ang = logtal U Mg = 59+9 h 58 + 9 / from (il M = q ~61-56 5M9?-11-5! -1-17 こ ویرا ht 5179 ? S. LAIR 9 5 stor $ 050)-() Stair