问题
当行列式中某两行或两列元素成比例时，该行列式会表现出来怎样的性质？选项
[A]. 该行列式不等于 $0$[B]. 该行列式等于 $1$
[C]. 该行列式等于 $0$
[D]. 该行列式不等于 $1$
该行列式等于 $0$
行列式中某两行或两列元素成比例时的性质（C001）
[B]. 该行列式等于 $1$
[C]. 该行列式等于 $0$
[D]. 该行列式不等于 $1$
标签： 线性代数基础
[B]. 该行列式等于 $1$
[C]. 该行列式等于 $0$
[D]. 该行列式不等于 $1$
行列式中某两行或两列元素相同时的性质（C001）
问题
当行列式中某两行或两列元素相同时，该行列式会表现出来怎样的性质？选项
[A]. 该行列式不等于 $0$[B]. 该行列式等于 $1$
[C]. 该行列式等于 $0$
[D]. 该行列式不等于 $1$
该行列式等于 $0$
行列式中某一行或列元素全为零时的性质（C001）
问题
当行列式中某一行或者某一列的元素全为 $0$ 的时，该行列式会表现出来怎样的性质？选项
[A]. 该行列式等于 $1$[B]. 该行列式等于 $0$
[C]. 该行列式不等于 $1$
[D]. 该行列式不等于 $0$
该行列式等于 $0$
行列式的可拆分性（C001）
问题
如果，行列式中某一行或者某一列的元素可以写成两数之和的形式，如：$\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$.
则，根据行列式的性质，可以对上面的行列式做什么样的转换？
选项
[A]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$ $\times$ $\left|\begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33}\end{array}\right|$[B]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$ $-$ $\left|\begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33}\end{array}\right|$
[C]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$ $+$ $\left|\begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33}\end{array}\right|$
[D]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} \frac{1}{a_{11}} & a_{12} & a_{13} \\ \frac{1}{a_{21}} & a_{22} & a_{23} \\ \frac{1}{a_{31}} & a_{32} & a_{33} \end{array}\right|$ $+$ $\left|\begin{array}{lll} \frac{1}{b_{11}} & a_{12} & a_{13} \\ \frac{1}{b_{21}} & a_{22} & a_{23} \\ \frac{1}{b_{31}} & a_{32} & a_{33}\end{array}\right|$
$\left|\begin{array}{lll} \textcolor{Red}{a_{11}} \textcolor{yellow}{+} \textcolor{cyan}{b_{11}} & a_{12} & a_{13} \\ \textcolor{Red}{a_{21}} \textcolor{yellow}{+} \textcolor{cyan}{b_{21}} & a_{22} & a_{23} \\ \textcolor{Red}{a_{31}} \textcolor{yellow}{+} \textcolor{cyan}{b_{31}} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} \textcolor{Red}{a_{11}} & a_{12} & a_{13} \\ \textcolor{Red}{a_{21}} & a_{22} & a_{23} \\ \textcolor{Red}{a_{31}} & a_{32} & a_{33} \end{array}\right|$ $\textcolor{yellow}{+}$ $\left|\begin{array}{lll} \textcolor{cyan}{b_{11}} & a_{12} & a_{13} \\ \textcolor{cyan}{b_{21}} & a_{22} & a_{23} \\ \textcolor{cyan}{b_{31}} & a_{32} & a_{33}\end{array}\right|$
常数公因子 $k$ 在行列式中的处理方式（C001）
问题
若行列式的某行或列有公因子 $k$, 则以下对该公因子的处理方式中，正确的是哪个？选项
[A]. $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ k a_{i 1} & k a_{i 2} & \cdots & k a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$ $=$ $\frac{1}{k}$ $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{i 1} & a_{i 2} & \cdots & a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right|$[B]. $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ k a_{i 1} & k a_{i 2} & \cdots & k a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$ $=$ $k$ $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{i 1} & a_{i 2} & \cdots & a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right|$
[C]. $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ k a_{i 1} & k a_{i 2} & \cdots & k a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$ $=$ $-k$ $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{i 1} & a_{i 2} & \cdots & a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right|$
[D]. $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ k a_{i 1} & k a_{i 2} & \cdots & k a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$ $=$ $k^{n}$ $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{i 1} & a_{i 2} & \cdots & a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right|$
$\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ \textcolor{red}{k} a_{i 1} & \textcolor{red}{k} a_{i 2} & \cdots & \textcolor{red}{k} a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$ $=$ $\textcolor{red}{k}$ $\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{i 1} & a_{i 2} & \cdots & a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right|$
行列式与转置行列式之间的关系（C001）
问题
已知，$D$ 为行列式，$D^{T}$ 为该行列式的转置行列式。则，$D$ 与 $D^{T}$ 之间的关系是什么？
选项
[A]. $D$ $=$ $D^{T}$[B]. $D$ $=$ $2 D^{T}$
[C]. $D$ $\neq$ $D^{T}$
[D]. $D$ $=$ $- D^{T}$
$D$ $=$ $D^{T}$