一道分块矩阵的坑题

nth233

A为二阶方阵，B为三阶方阵，|A| = |B| = 2，求

\begin{vmatrix} O& A \\ -2B & O \end{vmatrix}

nth233

\begin{aligned} &\begin{vmatrix} O& A^* \\ -2B & O \end{vmatrix} = (-1)^4 \cdot \det A^* \cdot \det(-2B) = \det(\det(A)A^{-1}) \cdot(-2)^3\det B \\ = & -8\cdot (\det A)^2 \cdot \frac{1}{\det A}\cdot \det B = -32 \end{aligned}

注意\displaystyle \begin{vmatrix} O& A^* \\ -2B & O \end{vmatrix} 只需列交换四次就能变成\displaystyle \begin{vmatrix} A^* & O \\ O &-2B \end{vmatrix}