Putnam 2021 B5

Say that an $n$-by-$n$ matrix $A = (a_{ij})_{1 \leq i,j \leq n}$ with integer entries is very odd if, for every nonempty subset $S$ of $\{1,2,\dots,n\}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \geq 1$.