求解分块矩阵的伴随矩阵

一、题目

已知 $\boldsymbol{A}, \boldsymbol{B}$ 为 $n$ 阶可逆矩阵, $\boldsymbol{E}$ 为 $n$ 阶单位矩阵，则伴随矩阵 $\left(\begin{array}{cc}\boldsymbol{A} & \boldsymbol{E} \\ \boldsymbol{O} & \boldsymbol{B}\end{array}\right)^{*}=?$

难度评级：

二、解析

已知：

$$
A^{*}=|A| A^{-1}
$$

因此：

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

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

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

$$
\textcolor{orange}{
\left[\begin{array}{ll}A & E \\ O & B\end{array}\right]^{*} = \left[\begin{array}{cc}|B| A^{*} & -A^{*} B^{*} \\ O & |A| B^{*}\end{array}\right]
}
$$

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