Is (2,0,1) an eigenvector of A?Let A = ( 5 1 2 0 6 0 1 − 1 4 ) (**) A ( 1 0 1 ) = ( 3 0 − 3 ) , A ( 1 1 0 ) = ( 6 6 0 ) , A ( 1 − 1 1 ) = ( 6 − 6 6 ) , ( 1 1 0 ) + ( 1 − 1 1 ) = ( 2 0 1 ) Which of the following is true?(1) (2,0,1) is an eigenvector associated to 3(2) (2,0,1) is an eigenvector associated to 6(3) (2,0,1) is not an eigenvector of AI found that only (2) is true by using the characteristic polynomial, but how do I prove that (2) is true using (**), as intended by the question?

Question

Is (2,0,1) an eigenvector of A?Let A =       (                            5                          1                          2                                      0                          6                          0                                      1                          −          1                          4                      )  (**)   A      (                            1                                      0                                      1                      )    =      (                            3                                      0                                      −          3                      )   ,   A      (                            1                                      1                                      0                      )    =      (                            6                                      6                                      0                      )  ,   A      (                            1                                      −          1                                      1                      )    =      (                            6                                      −          6                                      6                      )  ,          (                            1                                      1                                      0                      )    +      (                            1                                      −          1                                      1                      )    =      (                            2                                      0                                      1                      )  Which of the following is true?(1) (2,0,1) is an eigenvector associated to 3(2) (2,0,1) is an eigenvector associated to 6(3) (2,0,1) is not an eigenvector of AI found that only (2) is true by using the characteristic polynomial, but how do I prove that (2) is true using (**), as intended by the question?

Annabella Ferguson · Accepted Answer

Well, you have  A      (                            2                                      0                                      1                      )    =  A      (          (                                    1                                                1                                                0                              )        +          (                                    1                                                −            1                                                1                              )        )    =      (                            6                                      6                                      0                      )    +      (                            6                                      −          6                                      6                      )    =      (                            12                                      0                                      6                      )    =  6      (                            2                                      0                                      1                      )    .

Deborah Proctor · Accepted Answer

Note that if       v    1   and       v    2   are two eigenvectors of a matrix M with respect to the eigenvalue   λ, i.e.   M      v    1    =  λ      v    1   and   M      v    2    =  λ      v    2  , then any linear combination of       v    1   and       v    2  , is also an eigenvector of M corresponding to the eigenvalue   λ, since   M  (  a      v    1    +  b      v    2    )  =  a  M      v    1    +  b  M      v    2    =  a  λ      v    1    +  b  λ      v    2    =  λ  (  a      v    1    +  b      v    2    )Now for       v    1    =      (                            1                                      1                                      0                      )   and       v    2    =      (                            1                                      −          1                                      1                      )  , we have   A      v    1    =  6      v    1   and   A      v    2    =  6      v    2  . Hence       v    1   and       v    2   are eigenvectors of A corresponding to the eigenvalue 6. Hence       v    1    +      v    2    =      (                            2                                      0                                      1                      )   is an eigenvector of A corresponding to the eigenvalue 6

Let A=((5,1,2),(0,6,0),(1,-1,4)) (**) A=((1),(0),(1))=((3),(0),(-3)), A=((1),(1),(0))=((6),(6),(0)), A((1),(-1),(1))=((6),(-6),(6)), ((1),(1),(0))+((1),(-1),(1))=((2),(0),(1)) Which of the following is true?

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