Question

Q2. Consider the matrix A 6 3 0 -1 0-2 0 5 (a) Find all eigenvalues of the matrix A. (b) Find all eigenvectors of the matrix A. (c) Do you think that the set of the eigenvectors of A is a basis for the vector space R3? (Justify your answer

Answer 1

A= G 3 -7 O -1 0 - 2 S For Eigenvalue: IA-II = 0 6-7 3 - 1 -1- 0 5-1 (6-1) [(-:-) (5-1) - 0] -3 (0-0)-7(0-0) = 0 (6-4).(-:-1) (5-1) =0 pi 6--2-0 5-1=0 A-s Hence : Eigenvalus S Now finding Eigenvector corresponding to eigenvalue @ AEG A-AT = 6 - 6 3 O -6-1 - 2 S-6 3 7. 0 O

R2 + R and Rz ~) Rz + 3 3 R 3 -7 0 49 3 0 w/ this with o R3 - R3 R2 49 3 49 انر) R, 3 49 R2 O 1 0 0 R2 R, + 3 / 4 R2 o 0 O 0 0 O The system associated with abone matria 0 0 ET If we take x = t then Xi =t 22=0 13 - 0

+ Therefore t = 1 0 -BS A-XI = 3 - 7 -5-1 -2 5-S T 3 Rs) Prs - 1/2 R2 1 3 - 7 O 0 O R2 - R2 6 3 -7 1 o o 0

882 - 7 - 0 0 O The system associated with the above mania 0 X2 0 O “น we euce &z=t, 34 - It Uz = 0 x3=t > Therefore 2 let t=1 H = -1 A-II 6+1 3 - 1-) 51 3 7 C 0 C - 2

₂ R3 (interchanging) 7 3 -2 6 0 O R2 R2 - 2 and RA #IDO 1 3 7 O 1 -3 0 0 R, → R 7 R2 1 no 1 -3 0 0 The system associated with Cebone matria 2 0 24 22 - 3 0 0 0 น x3=t, then x = -2t 7 X2=3t 23 = t Therefore VA - mo 30

let t = 1 ( 23 - 1 Hence Eigenvalus 6 2 Eigenvector: Log Eigenvalues Eigenvector Eigenvalue -1, Eigenvector -4 non zero Eigen vectors all rows element prennt. So is 3. Che Rank of eigenvector eigenvector of O is a baris for Hence veetor spall R3

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