分块矩阵求逆法:下三角形式(C010)

问题

已知,$\boldsymbol{\textcolor{orange}{A}}$, $\boldsymbol{\textcolor{orange}{B}}$ 和 $\boldsymbol{\textcolor{orange}{C}}$ 是元素 的方阵,$\boldsymbol{\textcolor{orange}{O}}$ 是元素 的方阵
则,根据可逆矩阵的性质,$\textcolor{orange}{\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ \boldsymbol{B}^{-1} \boldsymbol{C} \boldsymbol{A}^{-1} & \boldsymbol{B}^{-1} \end{array}\right)$

[B].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ -\boldsymbol{A}^{-1} \boldsymbol{C} \boldsymbol{B}^{-1} & \boldsymbol{B}^{-1} \end{array}\right)$

[C].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ -\boldsymbol{B}^{-1} \boldsymbol{C} \boldsymbol{A}^{-1} & \boldsymbol{B}^{-1} \end{array}\right)$

[D].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ -\boldsymbol{B} \boldsymbol{C}^{-1} \boldsymbol{A} & \boldsymbol{B}^{-1} \end{array}\right)$


显示答案

$\left(\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}} & \boldsymbol{O} \\ \boldsymbol{\textcolor{yellow}{C}} & \boldsymbol{\textcolor{cyan}{B}} \end{array}\right)^{\textcolor{red}{-1}}$ $=$ $\left(\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} & \boldsymbol{O} \\ \textcolor{red}{-}\boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}} \boldsymbol{\textcolor{yellow}{C}} \boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} & \boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}} \end{array}\right)$