分块矩阵的逆运算和次方运算怎么算？

一、题目

已知 $A,B$ 均是 $n$ 阶可逆矩阵, 则以下错误的是哪个？

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

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

(C) ${\left[\begin{array}{ll}\mathit{A}& \mathit{O}\\ \mathit{O}& \mathit{B}\end{array}\right]}^n=\left[\begin{array}{cc}{\mathit{A}}^n& \mathit{O}\\ \mathit{O}& {\mathit{B}}^n\end{array}\right]$

(D) ${\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]}^n=\left[\begin{array}{cc}\mathit{O}& {\mathit{A}}^n\\ {\mathit{B}}^n& \mathit{O}\end{array}\right]$

难度评级：

二、解析

注意：本提问的是“错误的”是哪个，而不是“正确的”是哪个。

根据主对角线分块矩阵的逆运算法则和副对角线分块矩阵的逆运算法则，A、B、C 选项都是原有的公式，所以一定是正确的：

$${\left[\begin{array}{ll}A& O\\ O& B\end{array}\right]}^{-1}=\left[\begin{array}{cc}{A}^{-1}& O\\ O& B^{-1}\end{array}\right]$$

$${\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]}^{-1}=\left[\begin{array}{cc}\mathit{O}& {\mathit{B}}^{-1}\\ {\mathit{A}}^{-1}& \mathit{O}\end{array}\right]$$

$${\left[\begin{array}{ll}\mathit{A}& \mathit{O}\\ \mathit{O}& \mathit{B}\end{array}\right]}^n=\left[\begin{array}{cc}{\mathit{A}}^n& \mathit{O}\\ \mathit{O}& {\mathit{B}}^n\end{array}\right]$$

对于 D 选项，我们只需要令 $n=2$ 计算一下即可判断：

$${\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]}^2=$$

$$\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]=\left[\begin{array}{ll}\mathit{A}\mathit{B}& \mathit{O}\\ \mathit{O}& \mathit{B}\mathit{A}\end{array}\right]$$

事实上，当 $n$ 为偶数时：

$${\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]}^n=\left[\begin{array}{ll}\mathbf{(}\mathit{A}\mathit{B}{\mathbf{)}}^{\frac{\mathit{n}}{\mathbf{2}}}& \mathit{O}\\ \mathit{O}& \mathbf{(}\mathit{B}\mathit{A}{\mathbf{)}}^{\frac{\mathit{n}}{\mathbf{2}}}\end{array}\right]$$

当 $n$ 为奇数时：

$${\left[\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right]}^n=\left[\begin{array}{ll}\mathbf{(}\mathit{A}\mathit{B}{\mathbf{)}}^{\frac{\mathit{n}}{\mathbf{2}}}\mathit{A}& \mathit{O}\\ \mathit{O}& \mathbf{(}\mathit{B}\mathit{A}{\mathbf{)}}^{\frac{\mathit{n}}{\mathbf{2}}}\mathit{B}\end{array}\right]$$

综上可知，正确选项为 D.

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