Finding output with unique witness in matrix multiplication
You can reduce Boolean matrix multiplication (BMM) to this problem. (BMM is matrix multiplication over the OR/AND semiring with 0 and 1.) Imagine adding one more column to the first matrix A and one more row to the first matrix B, both of which are all-ones. If the BMM of A and B had a 0 in an entry, your new product over the integers will have 1, and if the BMM had a 1 in an entry, your new product over the integers will have at least a 2. Thus determining these &quot;exactly one&quot; entries is at least as hard as BMM.
Whether or not BMM can be solved faster than matrix multiplication over a ring or field is a major open problem.