解答题 （共 19 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$D=\left|\begin{array}{rrrr}
1 & 3 & -1 & 3 \\
3 & -2 & 2 & 4 \\
0 & 1 & 1 & -5 \\
1 & 4 & -2 & 3
\end{array}\right|$
计算
$$
D=\left|\begin{array}{cccc}
2 & -3 & 4 & 5 \\
3 & -2 & 3 & 4 \\
5 & 4 & -3 & 2 \\
4 & 6 & -4 & -5
\end{array}\right|
$$
计算 $D=\left|\begin{array}{llll}2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2\end{array}\right|$ ．
计算 $D=\left|\begin{array}{cccc}1+a & 1 & 1 & 1 \\ 1 & 1-a & 1 & 1 \\ 1 & 1 & 1+b & 1 \\ 1 & 1 & 1 & 1-b\end{array}\right|$ ．
计算 $n$ 阶行列式 $D=\left|\begin{array}{cccccc}a & b & 0 & \cdots & 0 & 0 \\ 0 & a & b & \cdots & 0 & 0 \\ 0 & 0 & a & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a & b \\ b & 0 & 0 & \cdots & 0 & a\end{array}\right|$ ．
计算 $n$ 阶行列式
$$
D_n=\left|\begin{array}{cccc}
a & b & \cdots & b \\
b & a & \cdots & b \\
\vdots & \vdots & & \vdots \\
b & b & \cdots & a
\end{array}\right|
$$
计算下列4阶行列式
$$
D=\left|\begin{array}{cccc}
1 & -1 & 1 & a-1 \\
1 & -1 & a+1 & -1 \\
1 & a-1 & 1 & -1 \\
a+1 & -1 & 1 & -1
\end{array}\right|
$$
计算 $n$ 阶行列式：
$$
D=\left|\begin{array}{ccccccc}
1 & 2 & 3 & \cdots & (n-2) & (n-1) & n \\
1 & -1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 2 & -2 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & (n-2) & -(n-2) & 0 \\
0 & 0 & 0 & \cdots & 0 & (n-1) & -(n-1)
\end{array}\right|
$$
计算行列式 $D=\left|\begin{array}{ccccc}0 & 1 & 1 & \cdots & 1 \\ 1 & 2 & 0 & \cdots & 0 \\ 1 & 0 & 3 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & 0 & 0 & \cdots & n\end{array}\right|$ ．
计算行列式
$$
D=\left|\begin{array}{ccccc}
x_1 & a_2 & a_3 & \cdots & a_n \\
a_1 & x_2 & a_3 & \cdots & a_n \\
a_1 & a_2 & x_3 & \cdots & a_n \\
\vdots & \vdots & \vdots & & \vdots \\
a_1 & a_2 & a_3 & \cdots & x_n
\end{array}\right|, x_i \neq a_i, i=1,2, \cdots, n
$$
计算 $D=\left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
2 & 2 & 3 & \cdots & n-1 & n \\
3 & 3 & 3 & \cdots & n-1 & n \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
n-1 & n-1 & n-1 & \cdots & n-1 & n \\
n & n & n & \cdots & n & n
\end{array}\right| .$
计算$D=\left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
x & 1 & 2 & \cdots & n-2 & n-1 \\
x & x & 1 & \cdots & n-3 & n-2 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
x & x & x & \cdots & 1 & 2 \\
x & x & x & \cdots & x & 1
\end{array}\right|,$
计算$D=\left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
2 & 2 & 3 & \cdots & n-1 & n \\
3 & 3 & 3 & \cdots & n-1 & n \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
n-1 & n-1 & n-1 & \cdots & n-1 & n \\
n & n & n & \cdots & n & n
\end{array}\right| .$
计算$D=\left|\begin{array}{cccccc}
1 & 2 & 3 & \cdots & n-1 & n \\
x & 1 & 2 & \cdots & n-2 & n-1 \\
x & x & 1 & \cdots & n-3 & n-2 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
x & x & x & \cdots & 1 & 2 \\
x & x & x & \cdots & x & 1
\end{array}\right|$
计算$D=\left|\begin{array}{llll}
x_1 & a_2 & a_3 & a_4 \\
a_1 & x_2 & a_3 & a_4 \\
a_1 & a_2 & x_3 & a_4 \\
a_1 & a_2 & a_3 & x_4
\end{array}\right|$ , 其中 $a_i \ne x_i$
计算$D=\left|\begin{array}{cccc}
x+a_1 & a_2 & \cdots & a_n \\
a_1 & x+a_2 & \cdots & a_n \\
\vdots & \vdots & & \vdots \\
a_1 & a_2 & \cdots & x+a_n
\end{array}\right|(x \neq 0)$ ,其中 $x \ne 0$
计算$D=\left|\begin{array}{ccc}
\frac{1}{3} & -\frac{5}{2} & \frac{2}{5} \\
3 & -12 & \frac{21}{5} \\
\frac{2}{3} & -\frac{9}{2} & \frac{4}{5}
\end{array}\right| .$
计算 $$
D=\left|\begin{array}{lll}
101 & 100 & 302 \\
198 & 200 & 603 \\
301 & 300 & 901
\end{array}\right|
$$
计算
$$
D=\left|\begin{array}{llll}
x_1 y_1 & x_1 y_2 & x_1 y_3 & x_1 y_4 \\
x_1 y_2 & x_2 y_2 & x_2 y_3 & x_2 y_4 \\
x_1 y_3 & x_2 y_3 & x_3 y_3 & x_3 y_4 \\
x_1 y_4 & x_2 y_4 & x_3 y_4 & x_4 y_4
\end{array}\right|
$$