Suppose $A$ is an $8 \times 8$ matrix over $\mathbb{Z}_5$ with the properties that:
- $per(A)=0$.
- The permanent of every $6 \times 6$ and $7 \times 7$ submatrix is zero.
My question is: from here, what must be true about $A$?
For example, one implication is that the polynomial $per(xI-A)$ has $0$ as a root with multiplicity at least $3$.
However, outside of this, nothing else (to me) seems clearly true about $A$. Can I, for instance, say anything about the invertibility of $A$?