单选题 （共 10 题 ），每题只有一个选项正确
四阶行列式 $\left|\begin{array}{cccc}a_1 & 0 & 0 & b_1 \\ 0 & a_2 & b_2 & 0 \\ 0 & b_3 & a_3 & 0 \\ b_4 & 0 & 0 & a_4\end{array}\right|$ 的值等于
$\text{A.}$ $a_1 a_2 a_3 a_4-b_1 b_2 b_3 b_4$.
$\text{B.}$ $a_1 a_2 a_3 a_4+b_1 b_2 b_3 b_4$.
$\text{C.}$ $\left(a_1 a_2-b_1 b_3\right)\left(a_3 a_4-b_3 b_4\right)$.
$\text{D.}$ $\left(a_2 a_3-b_2 b_3\right)\left(a_1 a_4-b_1 b_4\right)$.
行列式 $\left|\begin{array}{llll}0 & a & b & 0 \\ a & 0 & 0 & b \\ 0 & c & d & 0 \\ c & 0 & 0 & d\end{array}\right|=$
$\text{A.}$ $(a d-b c)^2$.
$\text{B.}$ $-(a d-b c)^2$.
$\text{C.}$ $a^2 d^2-b^2 c^2$.
$\text{D.}$ $b^2 c^2-a^2 d^2$.
若 $\alpha _1, \alpha _2, \alpha _3, \beta _1, \beta _2$ 都是 4 维列向量, 且 4 阶行列式 $\left| \alpha _1, \alpha _2, \alpha _3, \beta _1\right|=m$, $\left| \alpha _1, \alpha _2, \beta _2, \alpha _3\right|=n$, 则 4 阶行列式 $\left| \alpha _3, \alpha _2, \alpha _1, \beta _1+ \beta _2\right|$ 等于
$\text{A.}$ $m+n$.
$\text{B.}$ $-(m+n)$.
$\text{C.}$ $n-m$.
$\text{D.}$ $m-n$.
记行列式 $\left|\begin{array}{cccc}x-2 & x-1 & x-2 & x-3 \\ 2 x-2 & 2 x-1 & 2 x-2 & 2 x-3 \\ 3 x-3 & 3 x-2 & 4 x-5 & 3 x-5 \\ 4 x & 4 x-3 & 5 x-7 & 4 x-3\end{array}\right|$ 为 $f(x)$, 则方程 $f(x)=0$ 的的个数为
$\text{A.}$ 1.
$\text{B.}$ 2 .
$\text{C.}$ 3 .
$\text{D.}$ 4 .
设行列式 $D=\left|\begin{array}{cccc}3 & 0 & 4 & 0 \\ 2 & 2 & 2 & 2 \\ 0 & -7 & 0 & 0 \\ 5 & 3 & -2 & 2\end{array}\right|$, 则第四行各元素余子式之和的值为
$\text{A.}$ 14
$\text{B.}$ -14
$\text{C.}$ 28
$\text{D.}$ -28
$n$阶行列式
$$
\left|\begin{array}{cccccc}
a & b & 0 & \cdots & 0 & 0 \\
0 & a & b & \cdots & 0 & 0 \\
0 & 0 & a & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & a & b \\
b & 0 & 0 & \cdots & 0 & a
\end{array}\right|_{n \times n}
$$
得值为
$\text{A.}$ $a^n+(-1)^{n+1} b^n$
$\text{B.}$ 0
$\text{C.}$ $a^n-b^n$
$\text{D.}$ $a^n+b^n$
行列式
$$
\left|\begin{array}{cccc}
1 & -1 & 1 & x-1 \\
1 & -1 & x+1 & -1 \\
1 & x-1 & 1 & -1 \\
x+1 & -1 & 1 & -1
\end{array}\right|
$$
得值
$\text{A.}$ 0
$\text{B.}$ 1
$\text{C.}$ $x^4$
$\text{D.}$ $x^4-1$
行列式 $\left|\begin{array}{cccc}
\lambda & -1 & 0 & 0 \\
0 & \lambda & -1 & 0 \\
0 & 0 & \lambda & -1 \\
4 & 3 & 2 & \lambda+1
\end{array}\right|$
$\text{A.}$ $\lambda^4+\lambda^3+2 \lambda^2+3 \lambda+4 .$
$\text{B.}$ $\lambda^4-\lambda^3+2 \lambda^2-3 \lambda+4 .$
$\text{C.}$ 1
$\text{D.}$ 0
计算$D=\left|\begin{array}{ccccc}
1-a & a & 0 & 0 & 0 \\
-1 & 1-a & a & 0 & 0 \\
0 & -1 & 1-a & a & 0 \\
0 & 0 & -1 & 1-a & a \\
0 & 0 & 0 & -1 & 1-a
\end{array}\right|$
$\text{A.}$ $1+a+a^2+a^3+a^4+a^5$.
$\text{B.}$ $(1-a)^5$
$\text{C.}$ 0
$\text{D.}$ $D_5=1-a+a^2-a^3+a^4-a^5$.
行列式$\left|\begin{array}{cccc}
a & 0 & -1 & 1 \\
0 & a & 1 & -1 \\
-1 & 1 & a & 0 \\
1 & -1 & 0 & a
\end{array}\right|=$ 得值为
$\text{A.}$ $a^4$
$\text{B.}$ $a^2\left(a^2-4\right) .$
$\text{C.}$ $a^2\left(a^2+4\right) .$
$\text{D.}$ $a^4-1$