Regarding the answer in the following link:
Why is the trace of such a simplicial map is zero?
(The answer gives great background for the question).
I asked @John Palmieri a follow up question and he answered:
The $(i,j)$-entry of the matrix for $g_∗$ will be nonzero iff $g$ takes simplex $i$ to simplex $j$
How do you prove this claim?