Answer the following questions concerning the matrix $\boldsymbol{A}$ given by

$$A = \left( \begin{array}{ccc} 0 & 3 & 0 \\ -3 & 0 & 4 \\ 0 & -4 & 0 \end{array} \right)$$

In the following, $\boldsymbol{I}$ is the $3 \times 3$ identity matrix, $\boldsymbol{O}$ is the $3 \times 3$ zero matrix, $n$ is an integer greater than or equal to 0 and $t$ is a real number.

\begin{enumerate}
  \item Obtain all eigenvalues of the matrix $\boldsymbol{A}$.
  \item Find coefficients $a, b$ and $c$ of the following equation satisfied by $\boldsymbol{A}$:
$$\boldsymbol{A}^3 + a\boldsymbol{A}^2 + b\boldsymbol{A} + c\boldsymbol{I} = \boldsymbol{O}$$
  \item Obtain $\boldsymbol{A}^{2n+1}$.
  \item Since Equation (2) is satisfied, the following equation holds:
$$\exp(t\boldsymbol{A}) = p\boldsymbol{A}^2 + q\boldsymbol{A} + r\boldsymbol{I}$$
Express coefficients $p, q$ and $r$ in terms of $t$ without using the imaginary unit.
\end{enumerate}