Homework SourseFor the matrices A verify the given vectors X Y and Z are ei
For the matrices A, verify the given vectors, X, Y and Z, are eigenvectors and find a formula for A^n B, where B = [1, 1, 2]^T. a) [4 2 2 -1 1 -1 1 1 3] X = [1 0 -2] Y = [0 1 1] Z = [1 1 1] b) [2 -1 -1 2 5 1 -2 -1 3] X = [1 0 -1] Y = [2 1 1] Z = [1 1 0]![For the matrices A, verify the given vectors, X, Y and Z, are eigenvectors and find a formula for A^n B, where B = [1, 1, 2]^T. a) [4 2 2 -1 1 -1 1 1 3] X = [1](/WebImages/42/for-the-matrices-a-verify-the-given-vectors-x-y-and-z-are-ei-1129966-1761603425-0.webp "For the matrices A, verify the given vectors, X, Y and Z, are eigenvectors and find a formula for A^n B, where B = [1, 1, 2]^T. a) [4 2 2 -1 1 -1 1 1 3] X = [1")
Solution
(a) We have AX =(2,0,-4)T=2(1,0,-2)T=2X. Hence X is an eigenvalue of A with corresponding eigenvalue 2.
Also, AY = (0,2,2)T = 2(0,1,1)T = 2Y. Hence Y is an eigenvalue of A with corresponding eigenvalue 2.
Also, AZ = (4,4,4)T = 4(1,1,1)T = 4Z. Hence Z is an eigenvalue of A with corresponding eigenvalue 4.
(b) We have AX =(4,0,-4)T = 4(1,0,-1)T = 4X. Hence X is an eigenvalue of A with corresponding eigenvalue 4.
Also, AY = (4,2,2)T =2(2,1,1)T=2Y. Hence Y isan eigenvalue of A with corresponding eigenvalue 2.
Also, AZ = (4,4,0)T =4(1,1,0)T= 4Z. Hence Z is an eigenvalue of A with corresponding eigenvalue 4.
