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GATE 2022 CE (CE2) – Question 25

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple select

P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct?

  1. If P,Q invertible, $(PQ)^{-1}=Q^{-1}P^{-1}$
  2. If P,Q invertible, $(QP)^{-1}=P^{-1}Q^{-1}$
  3. If P,Q invertible, $(PQ)^{-1}=P^{-1}Q^{-1}$
  4. If P,Q not invertible, $(PQ)^{-1}=Q^{-1}P^{-1}$

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Show answer and explanation

Correct answer: (A) If P,Q invertible, $(PQ)^{-1}=Q^{-1}P^{-1}$; (B) If P,Q invertible, $(QP)^{-1}=P^{-1}Q^{-1}$

Explanation

The inverse of a product reverses multiplication order: $(PQ)^{-1}=Q^{-1}P^{-1}$. Verify by multiplying: $PQ Q^{-1}P^{-1}=I$.

Similarly $(QP)^{-1}=P^{-1}Q^{-1}$. In general matrices do not commute, and singular factors have no inverses. **Answer: A and B**.