JEE MAIN - Mathematics (2011 - No. 14)
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.
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.
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.
Explanation
$$\therefore$$ $$A' = A,B' = B$$
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.
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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