Powers of complex permutation matrix I am looking for $P^{100}$, where
$$
P=
        \begin{bmatrix}
        0 & i & 0 \\
        0 & 0 & i \\
        i & 0 & 0 \\
        \end{bmatrix}.
$$
The cycle repeats for $P^{13} = P$ and I find $P^{100}=P^{4}$. Is there a way to arrive at this answer quicker than by computing $P^2, P^3, \dots P^{13}$?

Yes, just compute $P^2$ and $P^3$, then you are done because of
$$
P^3=-\begin{pmatrix} i & 0 & 0 \cr 0 & i & 0 \cr 0 & 0 & i\end{pmatrix}.
$$
Of course, this matrix to the power of $4$ is the identity, and no smaller power, because no smaller power of $-i$ is equal to $1$.
