TITLE: There is no real $3\times3$ matrix $A$ for which $A^2=-I$.
QUESTION [5 upvotes]: I'm asked to prove or disprove this but I'm stuck.
I feel like this is actually true but i'm not sure as to how I can formulate a proof for this statement. Any leads?

REPLY [7 votes]: If $AA  = -I_3$, then
$\det(AA) = \det(-I_3) $
Does this help?