解答题 （共 6 题 ），解答过程应写出必要的文字说明、证明过程或演算步骤
计算
$$
\begin{aligned}
&D_n=\left|\begin{array}{ccccc}
\alpha+\beta & \alpha \beta & 0 & \cdots & 0 \\
1 & \alpha+\beta & \alpha \beta & \cdots & 0 \\
0 & 1 & \alpha+\beta & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & 0 & \cdots & \alpha+\beta
\end{array}\right|_{(n)}
\end{aligned}
$$
计算
$$
D_n=\left|\begin{array}{ccccc}
x & a & a & \cdots & a \\
-a & x & a & \cdots & a \\
-a & -a & x & \cdots & a \\
\cdots & \cdots & \cdots & \cdots & a \\
-a & -a & -a & \cdots & x
\end{array}\right|
$$
计算
$$
D=\left|\begin{array}{ccccc}
x+a_1 & a_2 & a_3 & \cdots & a_n \\
a_1 & x+a_2 & a_3 & \cdots & a_n \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
a_1 & a_2 & a_3 & \cdots & x+a_n
\end{array}\right|(x \neq 0)
$$
计算
$$
D_n=\left|\begin{array}{cccccc}
x+y & x y & 0 & \cdots & 0 & 0 \\
1 & x+y & x y & \cdots & 0 & 0 \\
0 & 1 & x+y & \cdots & 0 & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & 0 & \cdots & x+y & x y \\
0 & 0 & 0 & \cdots & 1 & x+y
\end{array}\right|
$$
求 $n$ 阶三对角线型行列式的值：
$$
D_n=\left|\begin{array}{ccccc}
2 & 1 & 0 & \cdots & 0 \\
1 & 2 & 1 & \cdots & 0 \\
0 & 1 & 2 & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & 0 & 0 & 2
\end{array}\right|
$$
求行列式的值：
$$
D_3=\left|\begin{array}{lll}
a_1 & a_2 & a_3 \\
a_3 & a_1 & a_2 \\
a_2 & a_3 & a_1
\end{array}\right|
$$