The GRE is this weekend and while working through a practice test I came across this question, and I couldn't see any clever way to work out the answer.
Of 2, 3, and 5, which are the eigenvalues of the following: \begin{bmatrix} 3 & 5 & 3 \\ 1 & 7 & 3\\ 1 &2 &8 \\ \end{bmatrix}
$A)$ None B)2, 3 C)2, 5 D)3, 5 E)2, 3, 5
I've found that often in these tests we can eliminate some of the answers based on the determinant or the trace, but for this one I'm stuck (the answer is C). Of course, I know that it could be the case that we actually have to work it out, I'm just hoping it isn't.
If you apply the determinantal definition of the characteristic polynomial, you can readily verify that 2 and 5 are eigenvalues (what do the resulting matrices look like?). Since you've indicated that you can eliminate (E) via the trace, this leaves (C) as the only option.
For example, the matrix for testing 2 is
$$ \left(\begin{matrix} 3 - 2 & 5 & 3 \\ 1 & 7 - 2 & 3 \\ 1 & 2 & 8 - 2 \\ \end{matrix}\right) = \left(\begin{matrix} 1 & 5 & 3 \\ 1 & 5 & 3 \\ 1 & 2 & 6 \\ \end{matrix}\right) $$