Prove or disprove:
if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$.
I'm pretty confident this is not true, but I've tried and tried to find a counter example without success. If someone contradicts this, I'd appreciate if you can outline your thought process when constructing the matrix $A$.
Also, another question in the same batch, asks:
Let $p(t)=t(t-0.25)(t-1)$ be the characteristic polynomial of $A^2$, is $A$ diagonalizable?
I'm thinking this is suppose to answer the previous question, assuming the answer here is false.
Here I know $A^2$ is diagonalizable, but I haven't made any substantial progress other than that.
I know I can go from the eigenvalues of $A$ to the eigenvalues of $A^k$, but not the other way around.