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]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} \frac{1}{a_{11}} & a_{12} & a_{13} \\ \frac{1}{a_{21}} & a_{22} & a_{23} \\ \frac{1}{a_{31}} & a_{32} & a_{33} \end{array}\right|$ $+$ $\left|\begin{array}{lll} \frac{1}{b_{11}} & a_{12} & a_{13} \\ \frac{1}{b_{21}} & a_{22} & a_{23} \\ \frac{1}{b_{31}} & a_{32} & a_{33}\end{array}\right|$[B]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$ $\times$ $\left|\begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33}\end{array}\right|$
[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]. $\left|\begin{array}{lll} a_{11}+b_{11} & a_{12} & a_{13} \\ a_{21}+b_{21} & a_{22} & a_{23} \\ a_{31}+b_{31} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$ $+$ $\left|\begin{array}{lll} b_{11} & a_{12} & a_{13} \\ b_{21} & a_{22} & a_{23} \\ b_{31} & a_{32} & a_{33}\end{array}\right|$
$\left|\begin{array}{lll} \textcolor{Red}{a_{11}} \textcolor{yellow}{+} \textcolor{cyan}{b_{11}} & a_{12} & a_{13} \\ \textcolor{Red}{a_{21}} \textcolor{yellow}{+} \textcolor{cyan}{b_{21}} & a_{22} & a_{23} \\ \textcolor{Red}{a_{31}} \textcolor{yellow}{+} \textcolor{cyan}{b_{31}} & a_{32} & a_{33} \end{array}\right|$ $=$ $\left|\begin{array}{lll} \textcolor{Red}{a_{11}} & a_{12} & a_{13} \\ \textcolor{Red}{a_{21}} & a_{22} & a_{23} \\ \textcolor{Red}{a_{31}} & a_{32} & a_{33} \end{array}\right|$ $\textcolor{yellow}{+}$ $\left|\begin{array}{lll} \textcolor{cyan}{b_{11}} & a_{12} & a_{13} \\ \textcolor{cyan}{b_{21}} & a_{22} & a_{23} \\ \textcolor{cyan}{b_{31}} & a_{32} & a_{33}\end{array}\right|$