1994 Putnam A4

Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such

that $A, A+B, A+2B, A+3B$, and $A+4B$ are all invertible matrices whose

inverses have integer entries. Show that $A+5B$ is invertible and that

its inverse has integer entries.

设 $A$ 和 $B$ 为 $2 \times 2$ 矩阵，其中包含整数项，例如

$A、A+B、A+2B、A+3B$ 和 $A+4B$ 都是可逆矩阵，其

逆有整数项。证明 $A+5B$ 是可逆的并且

它的逆元有整数项。