Question

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11. [5pts. Each] Let V = (1,-1,1), 12 =(2,1,3) and is =(-1,-1,2) be eigenvectors of the matrix A corresponding eigenvalue 4 = 2, =-2 and 1=3, respectively, and let v = (5.0,3) a. Express 1 as a linear combination of . 1, and is. b. Find Av.

Answer 1

Va'= (2,1,3) Vi' - (1,-1,1), ro' = (-1,-1,2) Here v' = (510,3) Let v' Can be written as are scalar. via Givil ! + 69 V3', where cor Ces ca = 4(1,-1,1)+(2,1,3)+(-1,2) → (5,0,3) C1 4+2C2-Cg, -G+CQ-Cg, 4+32 +263) D So from 0 a+2C₂-C3=5 -Gt-Cg 4+32 +2 3 = 3 0 ② +weget 32 -20325 Again + 4 we get 4G +63 = 3 fo So 5 t 2x © we get 36-263 +869 +249 25 2x3

» 12 all y Carl put (2=1 in 16 40+ C3 3 94x1 +69= 3 * C2 23-4 a good pat (2=1, C3=-1 in 3 -G+ G- C3 9 = -C3 = I-(1) mg = 2 now N'= avi 2 vit V2 V3 {Am ® Weg V'= 2V,' + vz! - v3' - Since v = (1,4,1), V2 = (2,1,3), V3 =(4,712) eigen - vectors of the matrix AA corresponding to the eigen -value 4,22 +2=-2, +3=3 respectivly

So -2 - Avi's divi's 2(1,-1, 1) = (2, -2, 2) 1:42] AV2'= da Va' -2(2,1,3) = (-4,-2,-6) {":12=-2 Avz' z dz vs' 3 (-1,-1, 2) = (-3,-3,6) [1323) SAV'A* 2x' +v-V3') A (2V, ') + AV2 - Avi = 2 AV,'+ AVR - AVE' 2(2,-2,2) + (-4,-2,-6) -(-3,-3,6) [by ①.6.0 262-2,2)+(-4, -2,-6) +(3,3,-6) =(4,-4,4)+(-4,-2,-6)+13, 3, -6) =( 4-4+3,-4-2 +3,4-6-6) Av =(3, -5,-8) [Am] a rote matin (d, a) in an eigen pair of then matria Aa= an