Trace zero matrix

Question

I have seen a problem where for a real matrix $A$ if $A^2=-I$ then trace $A$ is zero but I didn't understand why it is not true in general for a complex matrix $A.$

Answer 1

because a complex matrix can be: $$A= \begin{bmatrix} i&0\\ 0&i \end{bmatrix} $$ So: $$ A^2=\begin{bmatrix} i&0\\ 0&i \end{bmatrix} \begin{bmatrix} i&0\\ 0&i \end{bmatrix}= \begin{bmatrix} -1&0\\ 0&-1 \end{bmatrix}=-I $$

and note that this is true also for an $n \times n$ diagonal matrix that has all diagonal elements $a_{hh}=i$