I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now suppose for any finite-dimensional simplex $K$ I have the $i$-th Homology group with rational coefficients, $H_i(K,\mathbb{Q})$ and $f:K\rightarrow K$. Then we have the induced map $f_*:H_i(K,\mathbb{Q})\rightarrow H_i(K,\mathbb{Q})$. Since $f_*$ is a map from a finite-dimensional vetor space to itself, we can talk about its matrix and the alternating sum of the trace of the matrices of $i$-th Homology groups give the Lefschetz's Number for $f$.
1) I was wondering that what could be the $motivation/insight$ behind using the trace of the matrix of $f_*$ for various purposes.
2) Again, for any matrix, its trace as well as determinant are invariants, so I was wondering that are there any instances where we use the determinant of $f_*$ for associating something like Lefschetz's number. Is there any example in the literature where the determinant of $f_*$ serves any purpose or is this just a vague idea.
I tried to think about the 2nd question by taking some simplicial maps for lower-dimensional simplexes, but couldn't get anything promising. I searched about it on the internet, but couldn't find anything substantial.
Here is the Lefschetz fixed point theorem for finite sets: let $X$ be a finite set, and let $f : X \to X$ be an endomorphism of it. Let $k$ be a field of characteristic $0$ and let $k[f] : k[X] \to k[X]$ be the linear map induced by $f$ on the free $k$-vector space over $X$. Then $\text{tr}(k[f])$ is the number of fixed points of $f$.
(Already this highly degenerate version of the Lefschetz fixed point theorem can be surprisingly useful: for example, it can be used as part of a proof of Burnside's lemma.)
One way to conceptualize this argument is that there is an important sense in which taking the fixed points of $f$ is itself a kind of trace, and then the content of the claim is that the free vector space functor $k[-]$ "preserves traces." It is actually possible to prove the Lefschetz fixed point theorem itself along these lines; see, for example, Ponto and Shulman.
As for the determinant of $f_{\bullet}$, I'm not aware of any situation where that is taken. A related determinant can be used to define the Lefschetz zeta function.