For invertible matrices P and Q, which is the correct formula for the inverse of the product PQ?

For invertible matrices P and Q, which is the correct formula for the inverse of the product PQ?

• A. (PQ)^-1 = P^-1 Q^-1
• B. (PQ)^-1 = Q^-1 P^-1
• C. (PQ)^-1 = P Q
• D. (PQ)^-1 = P^-1 Q

Answer: (B)

When you invert a product of matrices, the order of the factors reverses. For invertible A and B, (AB)^{-1} = B^{-1}A^{-1}. Applying this to P and Q gives (PQ)^{-1} = Q^{-1}P^{-1}. You can verify by multiplication: (PQ)(Q^{-1}P^{-1}) = P(I)P^{-1} = I, and (Q^{-1}P^{-1})(PQ) = Q^{-1}(P^{-1}P)Q = Q^{-1}IQ = I, so Q^{-1}P^{-1} is indeed the inverse. The other expressions don’t reverse the order correctly, so they don’t cancel back to the identity in general.

When you invert a product of matrices, the order of the factors reverses. For invertible A and B, (AB)^{-1} = B^{-1}A^{-1}.

Applying this to P and Q gives (PQ)^{-1} = Q^{-1}P^{-1}. You can verify by multiplication: (PQ)(Q^{-1}P^{-1}) = P(I)P^{-1} = I, and (Q^{-1}P^{-1})(PQ) = Q^{-1}(P^{-1}P)Q = Q^{-1}IQ = I, so Q^{-1}P^{-1} is indeed the inverse.

The other expressions don’t reverse the order correctly, so they don’t cancel back to the identity in general.