Prove Matrix Correspondence is NP-complete
As you mentioned, we can reduce maximum clique problem to this. In fact we can reduce the more general subgraph isomorphism problem to this one easily. Given graphs $G$ and $H$, define $A$ and $B$ as their (undirected) incidence matrices. Then we can see that solving the matrix correspondence tells whether $G$ is isomorphic to some subgraph of $H$, which is an NP-complete problem.