$X_1=\left( \begin{array}{cc} 1 & 2 \\ -2 & 1 \end{array} \right)$，$X_2=\left( \begin{array}{cc} 6 & 5 \\ 1 & 3 \end{array} \right)$， \[ \begin{array}{r} X_n=\left( \begin{array}{cc} \displaystyle\frac{9}{4} & \displaystyle\frac{3}{2} \\ -\displaystyle\frac{1}{2} & \displaystyle\frac{1}{2} \end{array} \right)X_{n-1}-\left( \begin{array}{cc} \displaystyle\frac{5}{4} & \displaystyle\frac{3}{2} \\ -\displaystyle\frac{1}{2} & -\displaystyle\frac{1}{2} \end{array} \right)X_{n-2}+\left( \begin{array}{cc} \displaystyle\frac{1}{4} & \displaystyle\frac{3}{2} \\ -\displaystyle\frac{1}{2} & -\displaystyle\frac{3}{2} \end{array} \right) \\ (n=3,\ 4,\ 5,\ \cdots) \end{array} \] で定義される$2$次の正方行列の列がある．このとき，以下の問に答えよ．
(1) $A=\left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \end{array} \right)$，$B=\left( \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array} \right)$，$C=\left( \begin{array}{cc} \displaystyle\frac{5}{4} & \displaystyle\frac{3}{2} \\ -\displaystyle\frac{1}{2} & -\displaystyle\frac{1}{2} \end{array} \right)$，$P=\left( \begin{array}{cc} 2 & 3 \\ 1 & 1 \end{array} \right)$とする．$C=P^{-1}(kA+lB)P$を満たす実数$k$と$l$を求めよ．
(2) $C+C^2+\cdots +C^n=\left( \begin{array}{cc} \alpha_n & \beta_n \\ \gamma_n & \delta_n \end{array} \right) \ (n=1,\ 2,\ 3,\ \cdots)$とする．このとき，極限値$\displaystyle \lim_{n \to \infty}\alpha_n$，$\displaystyle \lim_{n \to \infty}\beta_n$，$\displaystyle \lim_{n \to \infty}\gamma_n$，$\displaystyle \lim_{n \to \infty}\delta_n$を求めよ．
(3) $X_n=\left( \begin{array}{cc} a_n & b_n \\ c_n & d_n \end{array} \right) \ (n=1,\ 2,\ 3,\ \cdots)$としたとき，極限値$\displaystyle \lim_{n \to \infty}a_n$，$\displaystyle \lim_{n \to \infty}b_n$，$\displaystyle \lim_{n \to \infty}c_n$，$\displaystyle \lim_{n \to \infty}d_n$が存在するかどうかを考察し，存在する場合はその値を求めよ．