I. Suppose that $\lambda$ is an eigenvalue of a regular matrix $\boldsymbol { P }$, prove that:

\begin{enumerate}
  \item $\lambda$ is not zero.
  \item $\lambda ^ { - 1 }$ is an eigenvalue of $\boldsymbol { P } ^ { - 1 }$ and $\lambda ^ { n }$ is an eigenvalue of $\boldsymbol { P } ^ { n }$, where $n$ is a positive integer.
\end{enumerate}

II. Suppose $\boldsymbol { P }$ is an orthogonal matrix. When the following symmetric matrix $\boldsymbol { A }$ can be diagonalized by $\boldsymbol { P }$, find the matrix $\boldsymbol { P }$ and obtain the diagonalized matrix.

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

III. When a matrix $\boldsymbol { P }$, and vectors $\boldsymbol { r }$ and $\boldsymbol { x }$ are given as

$$\boldsymbol { P } = \left( \begin{array} { c c c } 
1 & 1 & 1 \\
p & p ^ { 2 } & p ^ { 3 } \\
q & q ^ { 2 } & q ^ { 3 }
\end{array} \right) , \quad \boldsymbol { r } = \left( \begin{array} { c } 
r \\
r ^ { 2 } \\
r ^ { 3 }
\end{array} \right) , \quad \boldsymbol { x } = \left( \begin{array} { c } 
x \\
y \\
z
\end{array} \right) ,$$

where $p , q$, and $r$ are non-zero real numbers that differ from each other.

\begin{enumerate}
  \item Find the condition that $p$ and $q$ must satisfy in order for $\boldsymbol { P }$ to be a regular matrix.
  \item When $\boldsymbol { P } ^ { \mathrm { T } } \boldsymbol { x } = \boldsymbol { r }$ has a single solution, obtain $\boldsymbol { x }$. Here, $\boldsymbol { P } ^ { \mathrm { T } }$ is the transposed matrix of $\boldsymbol { P }$.
\end{enumerate}

IV. The matrix $\boldsymbol { P } _ { n }$ is an $n$-th order square matrix ( $n \geq 2$ ), as shown below, where $p$ and $q$ are real numbers that differ from each other.

$$\boldsymbol { P } _ { n } = \left( \begin{array} { c c c c c c } 
p + q & q & 0 & \cdots & 0 & 0 \\
p & p + q & \ddots & \ddots & \vdots & \vdots \\
0 & p & \ddots & \ddots & 0 & \vdots \\
\vdots & 0 & \ddots & \ddots & q & 0 \\
\vdots & \vdots & \ddots & \ddots & p + q & q \\
0 & 0 & \cdots & 0 & p & p + q
\end{array} \right)$$

\begin{enumerate}
  \item Obtain the recurrence formula satisfied by the determinant of $\boldsymbol { P } _ { n }$, $\left| \boldsymbol { P } _ { n } \right|$.
  \item Express the determinant $\left| \boldsymbol { P } _ { n } \right|$ in terms of $p , q$, and $n$, using the recurrence formula in Question IV.1.
\end{enumerate}