Abstract: There has been some recent interest toward understanding the prime ideals of various quantum groups. Useful results have comes from both representation theoretic and ring theoretic techniques. The algebra of quantum matrices has for various reasons formed an important test case for these approaches. In this talk, I present a third approach to quantum matrices: combinatorial. Specifically, I show how quantum matrices arise from considering paths in a certain grid graph. In fact, certain factor algebras can also be viewed this way. This viewpoint leads to new techniques one can use to understand an important class of prime ideals and to even slightly generalize some previously known results.