I read an answer on the Mathematica subforum that explained that a graph's adjacency matrix can be triangular (except the diagonal) if and only if the graph is a DAG. It is easy to see that the determinant of said matrix will always be 0.
Does this mean that every binary matrix (n x n) with a determinant of 0 will be an adjacency matrix to a DAG? If not, then can you provide a counterexample?
No, if determinant of adjacency matrix is 0 then its not always the case that the graph is a DAG. Simple example is following :
$$A=\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
On the other hand if the graph is DAG then the determinant of its adjacency matrix is always zero. To see this, first note that a directed graph is acyclic if and only if the vertices can be sorted in such a way that the adjacency matrix has upper triangular form with only zeros in the diagonal. So if A is the adjacency matrix of DAG and P is permuation matrix for the order of vertices then $det(PA)=0 \rightarrow det(A)det(P)=0 \rightarrow det(A)=0$