Question

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

Answer 1

Let, $A$ and $B$ are skew-symmetric matrices of the same order.

Then $A^T=-A,B^T=-B$........(1).

Now,

$(AB+BA{)}^T$

$=(AB{)}^T+(BA{)}^T$

$=B^T{A}^T+A^T{B}^T$ [ Using formula]

$=(-B)(-A)+(-A)(-B)$ [ Using (1)]

$=BA+AB$

$=AB+BA$ [ Since addition of matrices is commutative]

$(AB+BA{)}^T$$=AB+BA$.

So $(AB+BA)$ is symmetric.