Powers of matrices with all principal minors equal to 1

Consider a square matrix $A$ whose all principal minors are equal to $1$. Over a field, this property is inherited by any power of $A$, but this is not the case over an arbitrary commutative ring. We show that it is the case over any reduced ring, and also over the ring $\mathbb{Z}\mathrm{/}d$ for any integer $d$, and in some other settings (quotients of Prüfer domains and principal quotients of normal domains). This generalizes Problem B5 of the 2021 Putnam contest. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A={\left(a_{i,j}\right)}_{i,j\in \left[n\right]}$ is $1$-nullcyclic if all its diagonal entries are $1$ and if all the cyclic products $a_{i_1,i_2}{a}_{i_2,i_3}\cdots a_{i_k,i_1}$ with $k>1$ and distinct $i_1,i_2,\dots ,i_k$ vanish. We show that if $A$ is $1$-nullcyclic, then so is $A^m$ for any integer $m$. Furthermore, every $1$-nullcyclic matrix is $1$-principled over any commutative ring, while the converse holds over any reduced ring or any quotient of a ring by an integrally closed ideal. Along the way, we prove versions of these results that don’t require the diagonal entries to be $1$. Here, the principal minors of $A$ are required not to equal $1$ but rather to equal the respective products of diagonal entries of $A$. A crucial auxiliary result, which holds for any $n\times n$-matrix $A$, is that the cyclic products $a_{i_1,i_2}{a}_{i_2,i_3}\cdots a_{i_k,i_1}$ (with $k>1$ and distinct $i_1,i_2,\dots ,i_k$) are integral over the ideal generated by the principal minors of $A$ minus the corresponding products of diagonal entries of $A$.