1986 Putnam B6

Suppose $A,B,C,D$ are $n \times n$ matrices with entries in a field

$F$, satisfying the conditions that $AB^T$ and $CD^T$ are symmetric and

$AD^T - BC^T = I$. Here $I$ is the $n \times n$ identity matrix, and

if $M$ is an $n \times n$ matrix, $M^T$ is its transpose. Prove that

$A^T D - C^T B = I$.

假设 $A,B,C,D$ 是 $n \times n$ 矩阵，其中某个字段中有条目

$F$，满足$AB^T$和$CD^T$对称的条件

$AD^T - BC^T = I$。这里 $I$ 是 $n \times n$ 单位矩阵，并且

如果 $M$ 是 $n \times n$ 矩阵，则 $M^T$ 是它的转置。证明

$A^T D - C^T B = I$。