\textbf{Problem 2}

\textbf{I.} Answer the following questions about the matrix $\boldsymbol { P }$:
$$\boldsymbol { P } = \left( \begin{array} { c c c } 
0 & 0 & \frac { 3 } { 2 } \\
2 & 0 & 0 \\
0 & \frac { 1 } { 3 } & 0
\end{array} \right)$$
\begin{enumerate}
  \item Obtain all eigenvalues of the matrix $\boldsymbol { P }$ and the corresponding eigenvectors with unit norms.
  \item Obtain $P ^ { 2 }$ and $P ^ { 3 }$.
\end{enumerate}

\textbf{II.} Let $\boldsymbol { A }$ be the real matrix given by the block diagonal matrix:
$$\boldsymbol { A } = \left( \begin{array} { c c c c c } 
0 & 0 & c & 0 & 0 \\
a & 0 & 0 & 0 & 0 \\
0 & b & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & e \\
0 & 0 & 0 & d & 0
\end{array} \right)$$
Express succinctly the necessary and sufficient condition on $a , b , c , d$, and $e$, such that there exists a positive integer $m$ for which $\boldsymbol { A } ^ { m }$ is the identity matrix (proof is not required).

\textbf{III.} The matrix $M$ is a square matrix of order 12 with all elements taking either 0 or 1, such that each row and column has exactly one element being 1. Let $k _ { 0 }$ be the minimum value of the positive integer $k$ such that $M ^ { k }$ is the identity matrix. For all possible matrices $M$, give the maximum value of $k _ { 0 }$ (proof is not required).