If $A$ and $B$ have the same eigenvalues...

Question

I'm currently studying for my Linear Algebra final tomorrow, and this question irked me:

If $A$ and $B$ have the same eigenvalues, then
(a) $A$ and $B$ must be similar matrices.
(b) $A$ and $B$ must have the same eigenvectors.
(c) $\text{det} A = \text{det} B$
(d) (a) and (b)
(e) all of the above

I ruled out (b), and thus (d), easily. It turns out the right answer was (c), but I came up with counterexamples that proved (c) wrong.

When the question says "same eigenvalues," I started by thinking about matrices with different algebraic multiplicities for the same eigenvalues. Say $4 \times 4$ matrix $A$ has eigenvalues 1, 1, 1, 2 and $4 \times 4$ matrix $B$ has eigenvalues 1, 1, 2, 2. They certainly have the same eigenvalues, but they're certainly not similar. And their determinants are, obviously, different.

Then I considered matrices with the same eigenvalues and the same algebraic multiplicites. But then, both (a) and (c) must be right, which isn't an option.

I guess I'm missing something here?

Answer 1

None of them is actually true, though changing the first line to

• If $A$ and $B$ have the same eigenvalues with the same algebraic multiplicities

makes (c) true. Even adding this does not make (a) true: for example, $$A=\pmatrix{2&0\cr0&2\cr}\quad\hbox{and}\quad B=\pmatrix{2&1\cr0&2\cr}$$ are not similar. The matrix $B$ is an example of a Jordan matrix, don't know whether or not this is on your course.