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：

$| \begin{array}{lll} a_{11} + b_{11} & a_{12} & a_{13} \\ a_{21} + b_{21} & a_{22} & a_{23} \\ a_{31} + b_{31} & a_{32} & a_{33} \end{array} |$ .

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].

| \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} |

=

| \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} |

+

| \begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33} \end{array} |

[B].

| \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} |

=

| \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} |

+

| \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} |

[C].

| \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} |

=

| \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} |

\times

| \begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33} \end{array} |

[D].

| \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} |

=

| \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} |

-

| \begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33} \end{array} |

答案

$| \begin{array}{lll} a_{11} + b_{11} & a_{12} & a_{13} \\ a_{21} + b_{21} & a_{22} & a_{23} \\ a_{31} + b_{31} & a_{32} & a_{33} \end{array} |$ $=$ $| \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} |$ $+$ $| \begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33} \end{array} |$

问题

选项

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