I have a real square matrix $A$. I am told to prove that there is $A$ such that $A^2+2A+5I=0$ if and only if $n$ is even. (If $A$ is 6x6, $n=6$)
I honestly have no clue how to start. Maybe I could turn this into a question with minimal polynomial and use $x^2+2x+5$. This polynomial doesn't have a root, and it is making me even more confused. Could someone help? Thank you.