Graph planarity testing via adjacency matrix
Ignoring trivial responses like, recreate the graph from the matrix and apply any standard planarity algorithm, the closest I know of to a matrix-based planarity test is Whitney's planarity criterion. But it uses the incidence matrix, not the adjacency matrix, and is a mathematical criterion rather than a polynomial time algorithm.

It states that a graph is planar if and only if its matroid (the independence matroid of the columns of the signed incidence matrix) has a dual (the matroid for the dual space of the column space) which is also a graphic matroid.