The product of matrices (PQ)-1P is A. P-1 B. Q-1 C. P-1Q-1 P D. PQ P-1

The correct answer is $\boxed{\text{C. }P^{-1}Q^{-1}P}$.

Let $A$, $B$, and $C$ be matrices. Then, the following properties hold:

• $(AB)^{-1} = B^{-1}A^{-1}$
• $(A^{-1})^{-1} = A$
• $(AB)C = A(BC)$
• $A(B+C) = AB + AC$
• $(A+B)C = AC + BC$
• $A(kB) = k(AB)$
• $(kA)B = A(kB)$
• $A^2 = AA$
• $A(A^{-1}) = I$
• $(A^{-1})A = I$

In this case, we have $A = PQ$ and $B = P$. Substituting into the first property, we get $(PQ)^{-1}P = P^{-1}(PQ)^{-1}$. Applying the second property, we get $P^{-1}(PQ)^{-1} = (P^{-1}PQ)^{-1}$. Applying the third property, we get $(P^{-1}PQ)^{-1} = P^{-1}Q^{-1}$. Therefore, the product of matrices $(PQ)^{-1}P$ is $P^{-1}Q^{-1}P$.

Here is a brief explanation of each option:

• Option A: $P^{-1}$. This is not the correct answer because $P^{-1}$ is not the product of matrices $(PQ)^{-1}P$.
• Option B: $Q^{-1}$. This is not the correct answer because $Q^{-1}$ is not the product of matrices $(PQ)^{-1}P$.
• Option C: $P^{-1}Q^{-1}P$. This is the correct answer because it is the product of matrices $(PQ)^{-1}P$.
• Option D: $PQ P^{-1}$. This is not the correct answer because $PQ P^{-1}$ is not the product of matrices $(PQ)^{-1}P$.