\section*{Problem 2}
Answer the following questions about the square matrix $A$ of order 3:

$$A = \left( \begin{array} { c c c } 
3 & 0 & 1 \\
- 1 & 2 & - 1 \\
- 2 & - 2 & 1
\end{array} \right)$$

I. Find all eigenvalues of $A$.\\
II. Find the matrix $A ^ { n }$, where $n$ is a natural number.\\
III. The square matrix $\boldsymbol { B }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A }$. Prove that any eigenvector $\boldsymbol { p }$ of $\boldsymbol { A }$ is also an eigenvector of $\boldsymbol { B }$.\\
IV. Find the square matrix $\boldsymbol { B }$ of order 3 that meets $\boldsymbol { B } ^ { 2 } = \boldsymbol { A }$, where $\boldsymbol { B }$ is diagonalizable and all eigenvalues of $\boldsymbol { B }$ are positive.\\
V. The square matrix $\boldsymbol { X }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { X } = \boldsymbol { X } \boldsymbol { A }$. When $\operatorname { tr } ( \boldsymbol { A } \boldsymbol { X } ) = d$, find the maximum of $\operatorname { det } ( \boldsymbol { A } \boldsymbol { X } )$ as a function of $d$.

Here, $d$ is positive real and all eigenvalues of $X$ are positive. In addition, $\operatorname { tr } ( M )$ is the trace (the sum of the main diagonal elements) of the square matrix $\boldsymbol { M }$, and $\operatorname { det } ( \boldsymbol { M } )$ is the determinant of the matrix $\boldsymbol { M }$.