Question:
The characteristic polynomial of a square matrix B is given as $t^2(t^2+1)$. From this information, deduce all the possible values of rank($B^2$)
Here is my attempt so far:
The degree of the characteristic polynomial p(t)=4. The eigenvalues are 0, $\pm i $. The rank corresponds to the number of non-zero eigenvalues.
My confusion starts here, since the eigenvalues are complex.
I would appreciate any help with this question. Thank you.
Well in general it holds that rank$(AB) \leq$ min$($rank$(A), rank(B)$) so in your case we get:
rank$(A^2) \le $rank$(A)$. From here it quickly follows that rank$(A)=1$ or $2.$
From here you can probably figure out the rest.