QJEEM11Ma73f2: JEE Main 2011 Mathematics - Class 12 JEEM11Ma73f2 Solution: Let and be two symmetric matrices of order . Statement - 1 :…

Question

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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

with

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.

Accepted Answer

\therefore

A' = A,B' = B

Now

\,\,\,\left( {A\left( {BA} \right)} \right)' = \left( {BA} \right)'A'

= \left( {A'B'} \right)A' = \left( {AB} \right)A = A\left( {BA} \right)

Similarly

\left( {\left( {AB} \right)A} \right)' = \left( {AB} \right)A

So,

A\left( {BA} \right)\,\,\,\,

and

A\left( {BA} \right)\,\,\,\,

are symmetric matrices.

Again

\left( {AB} \right)' = B'A' = BA

Now if

BA=AB

, then

AB

is symmetric matrix.

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