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].

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ k a_{i 1} & k a_{i 2} & \dots & k a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

=

k

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

[B].

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ k a_{i 1} & k a_{i 2} & \dots & k a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

=

- k

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

[C].

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ k a_{i 1} & k a_{i 2} & \dots & k a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

=

k^{n}

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

[D].

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ k a_{i 1} & k a_{i 2} & \dots & k a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

=

\frac{1}{k}

| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |

答案

$| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ k a_{i 1} & k a_{i 2} & \dots & k a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |$ $=$ $k$ $| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ \dots & \dots & \dots & \dots \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array} |$

问题

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