Question

if A and B are symmetric matrix then AB+BA
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Answer 1

Answer:

Step-by-step explanation:

We have,

$\sf{A\,\,\&\,\,B\,\,are\,\,symmetric\,\,matrix}$

So,

$\rm{A^{\prime}=A\,\,\,\,\&\,\,\,\,B^{\prime}=B\,\,\,\,\,\,\,\,\,\,\,\,...(1)}$

Now,

$\rm{\left(AB+BA\right)^{\prime}=\left(AB\right)^{\prime}+\left(BA\right)^{\prime}}$

$\rm{\implies\left(AB+BA\right)^{\prime}=\left(B\right)^{\prime}\cdot\left(A\right)^{\prime}+\left(A\right)^{\prime}\cdot\left(B\right)^{\prime}}$

From (1), we get,

$\rm{\implies\left(AB+BA\right)^{\prime}=B\cdot\,A+A\cdot\,B}$

$\rm{\implies\left(AB+BA\right)^{\prime}=BA+AB}$

$\rm{\implies\left(AB+BA\right)^{\prime}=AB+BA}$

Hence (AB+BA) is a symmetric matrix