# 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. Option 1) Statement-1 is true, Statement-2 is false. Option 2) Statement-1 is false, Statement-2 is true. Option 3) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 . Option 4) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

N neha
P Prateek Shrivastava

As we learnt in

Property of Transpose Conjugate -

$\left ( AB \right )^{\Theta }=B^{\Theta } A^{\Theta }$

- wherein

$A^{\Theta }$ is the conjugate matrix of $A$

Given that $A^{T}=A\, \, \, and\, \, \,B^{T}=B$

Then $\left [ A\left ( BA\right ) \right ]^{T}= \left ( BA \right )^{T}A^{T}= A^{T}B^{T}A^{T}= ABA$

Similarly $\left [\left ( AB\right )A \right ]^{T}=ABA$

AB is symmmetric matrix.

If AB is commulative.

Option 1)

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

Incorrect Option

Option 2)

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

Incorrect Option

Option 3)

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

Incorrect Option

Option 4)

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

Correct Option

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