Matrices — JEE Main · 2011 · 10 Apr 2011 · Offline Question with Solution

Let A and B be two symmetric matrices of order 3.

Statement-1: A(BA) and (AB)A are symmetric matrices.

Statement-2: AB is 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✓ Correct
(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

Answer: (A)

Solution

With

A

and

B

symmetric,

(A (B A))^{T} = (B A)^{T} A^{T} = (A^{T} B^{T}) A = A B A = A (B A)

and likewise

((A B) A)^{T} = A^{T} (A B)^{T} = A (B^{T} A^{T}) = A (B A) = (A B) A

, so both

A (B A)

and

(A B) A

are symmetric and the first statement is true. The second is also true, since if

A B = B A

then

(A B)^{T} = B^{T} A^{T} = B A = A B

. But the symmetry of

A (B A)

and

(A B) A

holds whether or not

A

and

B

commute, so the second statement is not the correct explanation of the first.

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option

JEE Main · 2011 · 10 Apr 2011 · Offline · Matrices

Solution

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