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

Unlock all questions

This demo includes only 20 questions. Upgrade to access hundreds of questions, flashcards, exam simulations, and disable ads.

Full question bankExam simulationsFlashcards
From $9.99Unlock all

Boost your skills in A Level Further Mathematics Core Pure. Study with flashcards and multiple choice questions, each question is followed by hints and explanations. Get prepared for your exam with confidence!

Multiple Choice

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

Explanation:
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.

More Multiple Choice Questions
Subscribe

Get the latest from Examzify

You can unsubscribe at any time. Read our privacy policy