We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new explanation avoids linear-algebraic arguments and instead exploits a classical connection between subgraph and homomorphism counts.
On random $\pm 1$ matrices: Singularity and Determinant
Condensation of Determinants
On determinants of tournaments and $\mathcal{D}_k$
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