Step-by-Step Solution
Key Concept: For a matrix product, $|AB| = |A||B|$ and the trace relates to cofactor operations
Given $|P| = -5$ and $|Z| = 10$. Since $Z = PQ^{-1}$, we have $|Z| = \frac{|P|}{|Q|}$, so $10 = \frac{-5}{|Q|}$, giving $|Q| = -\frac{1}{2}$. Then $Z[Q] = P·adj(Q)$, and $\text{Tr}(adj(Q)) = \text{Tr}(P·adj(Q)) = (\frac{-1}{2})\text{Tr}(Z) = -1$.
Correct Answer: -1