Existence of a real $4 \times 4$ matrix satisfying the equation $x^2+1=0$. Does there exist a real $4 \times 4$ matrix satisfying the equation $x^2+1=0$?
I have tried using determinants to see if I can arrive at some sort of contradiction, but it doesn't quite help.
Also, I have tried putting entries ($a,b,c,d \ldots$ etc.) in the matrix to see if it gets me anywhere, and it doesn't. And, companion matrices don't help either because the polynomial is of degree $2$ while the matrix is a real $4 \times 4$ matrix.
Is there some sort of approach that I am definitely not aware of at this point?

This works:
$$\begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix}^2=-I_4$$
