Prove that a topological sort exists for a given graph using only linear algebraic properties of the adjacency matrix or the laplacian.

Or even, give a proof that a graph is a DAG iff there exists a topological sort using only linear algebraic operations on the adjacency matrix.

The point is, I'm sure you can do this; in some sense, we can perform every operation on a graph as we can its adjacency matrix, but why make it so complicated when there are easier pictures to work with?

Why perform surgery with a chainsaw when a scalpel will do without such a mess?