Question

If A and B are symmetric matrices of same order, prove that AB- BA is a 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 $B^{\prime }\to B$

$=BA-AB=-(AB-BA)$ (by taking negative common)

we know that a matrix is said to b skew symmetric matrix if $A=-A$

Hence $AB-BA$ is skew symmetric matrices