Question

If $A$ and $B$ are skew symmetric matrices of same order then

• A. $AB$ is skew symmetric
• B. $AB+BA$ is symmetic
• C. $AB-BA$ is symmetric
• D. none of these

Answer 1

The correct option is B $AB+BA$ is symmetic

$A$ and $B$ are skew symmetric matrix

$A^T=-A{B}^T=-B$

$(AB{)}^T=B^T{A}^T=(-B\left)\right(-A)=BA$

$(BA{)}^T=A^T{B}^T=(-A\left)\right(-B)=AB$

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

$=-(AB-BA)$

$\therefore AB-BA$ is skew symmetric

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

$=AB+BA$

$\therefore AB+BA$ is symmetric