例 2.70

依赖于

• 例 2.69

被以下题目直接调用

• 无

例 2.70

设 $A,B,C,D$ 都是 $n$ 阶矩阵，求证：

$$\begin{array}{rl}|M|& =\left|\begin{array}{cccc}A& B& C& D\\ B& A& D& C\\ C& D& A& B\\ D& C& B& A\end{array}\right|\\ & =|A+B+C+D||A+B-C-D||A-B+C-D||A-B-C+D|.\end{array}$$

解答

证明 反复利用例 2.69 的结论可得

$$\begin{array}{rl}|M|& =\left|\left(\begin{array}{cc}A& B\\ B& A\end{array}\right)+\left(\begin{array}{cc}C& D\\ D& C\end{array}\right)\right|\cdot \left|\left(\begin{array}{cc}A& B\\ B& A\end{array}\right)-\left(\begin{array}{cc}C& D\\ D& C\end{array}\right)\right|\\ & =\left|\begin{array}{cc}A+C& B+D\\ B+D& A+C\end{array}\right|\cdot \left|\begin{array}{cc}A-C& B-D\\ B-D& A-C\end{array}\right|\\ & =|A+B+C+D||A-B+C-D||A+B-C-D||A-B-C+D|.\end{array}$$