Question

If A and B are symmetric matrices of same order, prove that AB+BA is a skew-symmetric matrix.

Answer 1

Given :$A$ and $B$ are symmetric matrices.

$\Rightarrow A=A^{\prime }$ and $B=B^{\prime }$

From the property of transpose of matrices. we have

$AB=BA$

Now consider $AB-BA$ and by taking transpose of it, we get

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

Replace $A^{\prime }\to -A$ and $B^{\prime }\to -B$

$=BA+AB=(AB+BA)$

we know that a matrix is said to be skew symmetric matrix if $A=-A$ and $B=-B$

Hence $AB+BA$ is skew symmetric matrices