Let A and B be two symmetric matrices of order 3 .Statement-1 : ABA and AB^2A are symmetric matrices.Statement-2 : AB is a symmetric matrix if matrix - Mathematics MCQ FIITJEE AIEEE 2011 (Set Q) 2026 | ExamTest.live

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Let A and B be two symmetric matrices of order 3 .Statement-1 : A B A and A B^{2} A are symmetric matrices.Statement-2 : A B is a symmetric matrix if matrix multiplication of A and B is commutative.

A $Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1$
B $Statement-1 is true, Statement-2 is false.$
C $Statement-1 is false, Statement-2 is true.$
D $Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1$

Solution & Step-by-step Explanation

Given A^{T} = A and B^{T} = B .Statement 1: (A B A)^{T} = A^{T} B^{T} A^{T} = A B A . (Symmetric) (A B^{2} A)^{T} = A^{T} (B^{T})^{2} A^{T} = A B^{2} A . (Symmetric)Statement 1 is true.Statement 2: (A B)^{T} = B^{T} A^{T} = B A .For A B to be symmetric, (A B)^{T} = A B ⟹ B A = A B .Statement 2 is true.Statement 2 is not the explanation for Statement 1 because Statement 1 is true regardless of whether A and B commute.

Try it yourself before checking the explanation above.

Let A and B be two symmetric matrices of order 3 .Statement-1 : A B A and A B^{2} A are symmetric matrices.Statement-2 : A B is a symmetric matrix if matrix multiplication of A and B is commutative.

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

Statement-1 is true, Statement-2 is false.

Statement-1 is false, Statement-2 is true.

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

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