GATE CE 2022 Set 2 — Question 25
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Engineering Mathematics → Linear Algebra → Matrix Algebra & Operations
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Question
and are two square matrices of the same order. Which of the following statement(s) is/are correct?
Correct answer
(A) If P and Q are invertible, then [PQ]⁻¹ = Q⁻¹P⁻¹.; (B) If P and Q are invertible, then [QP]⁻¹ = P⁻¹Q⁻¹.
Solution
The inverse of a product of two invertible matrices and is given by the property .
1.Applying this to the product , we get . Thus, statement (A) is correct.
2.Applying this to the product , we get . Thus, statement (B) is correct.
3.Statement (C) is generally incorrect because matrix multiplication is non-commutative ( in general).
4.Statement (D) is incorrect because if or is not invertible, their product is also not invertible, and the inverse does not exist.
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