A $3×3$ real symmetric matrix $M$ admits $(1, 2, 3)^T$ and $(1, 1,−1)^T$ as eigenvectors. Which of the following(s) is/are surely an eigenvector of $M$?
- (a) $(1,−1, 0)^T$
- (b) $(−5, 1, 1)^T$
- (c) $(3, 2, 1)^T$
- (d) None
Answer: (d)
I have tried to solve it. I know eigenvectors of a real symmetric matrix corresponding to different eigenvalues are orthogonal to each other. (But it's not true in general if we take two eigenvectors corresponding to the same eigenvalue). ($1$)
Now notice that the vectors given in the questions are orthogonal to each other. But none of (a), (b), (c) is orthogonal to both of the given vectors. ($2$)
My claim is vectors in options (a), (b) and (c) may not be an eigenvector of a symmetric matrix having eigenvectors as given in the question. So, I want to construct a real symmetric matrix having two eigenvectors $(1, 2, 3)^T$ and $(1, 1,−1)^T$ but not having $(1,−1, 0)^T$ (or, $(−5, 1, 1)^T$ or $(3, 2, 1)^T$) as eigenvector using the facts ($1$)and ($2$).
But how to construct such matrices? Can I get any method of constructing such real symmetric matrices? Or is there any other argument to answer it in short? Thanks for the assistance in advance.