$(\boldsymbol{A}+\boldsymbol{B})^{-1}$ 是否等于 $\boldsymbol{A}^{-1}$ $+$ $\boldsymbol{B}^{-1}$ ?(C010)

问题

根据可逆矩阵的性质,一般情况下,$\textcolor{orange}{(\boldsymbol{A}+\boldsymbol{B})^{-1}}$ $\textcolor{orange}{\boldsymbol{A}^{-1}}$ $\textcolor{orange}{+}$ $\textcolor{orange}{\boldsymbol{B}^{-1}}$ $?$

选项

[A].   $(\boldsymbol{A}+\boldsymbol{B})^{-1}$ $=$ $\boldsymbol{A}^{-1}$ $+$ $\boldsymbol{B}^{-1}$

[B].   $(\boldsymbol{A}+\boldsymbol{B})^{-1}$ $\neq$ $\boldsymbol{A}^{-1}$ $+$ $\boldsymbol{B}^{-1}$


显示答案

$(\boldsymbol{\textcolor{orange}{A}}+\boldsymbol{\textcolor{orange}{B}})^{\textcolor{cyan}{-1}}$ $\textcolor{red}{\neq}$ $\boldsymbol{\textcolor{orange}{A}}^{\textcolor{cyan}{-1}}$ $+$ $\boldsymbol{\textcolor{orange}{B}}^{\textcolor{cyan}{-1}}$