If A and B are symmetric matrices of same order, then show that AB is symmetric if and only if A and

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3 and 4 .Determinants and Matrices

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If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB = BA$.

Option A

Option B

Option C

Option D

Solution

Solution since $A$ and $B$ are both symmetric matrices, therefore $A^{\prime}=A$ and $B^{\prime}=B$.

Let $AB$ be symmetric, then $(A B)^{\prime}=A B$

But $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}=\mathrm{BA}(\text { Why } ?)$

Therefore $ B A=A B$

Conversely, if $\mathrm{AB}=\mathrm{BA}$, then we shall show that $\mathrm{AB}$ is symmetric.

Now $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}$

$=\mathrm{B} \mathrm{A}(\text { as } \mathrm{A} \text { and } \mathrm{B}$ are symmetric)

$=\mathrm{AB}$

Hence $AB$ is symmetric.

Standard 12

Mathematics

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