Find matrix $A$ such that $A^2 = -I$

Question

I'm trying to solve for matrix $A$ where $A^2 = - I$, where $I$ is the identity matrix of the same order as $A$. Also, my second question: is there a matrix $A$ where $A^3 = 0$, but $A^2 \neq 0$? Please advise, thanks.

Answer 1

How about $iI$?

For the second, a nilpotent matrix like $\begin{pmatrix}0&1&1\\0&0&1\\0&0&0\end{pmatrix}$ does the trick.

Answer 2

What's the square of $\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}$?

What are $\begin{pmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{pmatrix}^2$ and $\begin{pmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{pmatrix}^3$?