Determine whether a system of $n$ linear equations has solutions in $\{0, 1\}^n$ in polynomial time
1-IN-3SAT is the following problem: given a collection of 3-clauses (just like 3SAT), decide whether there is a truth assignment which satisfies exactly one literal in each clause. 1-IN-3SAT is known to be NP-complete, see for example here. You can reduce 1-IN-3SAT to your problem – I'll let you work out the details.