Question

If $A$ and $B$ are symmetric matrices, then $AB-BA$ is a

• A. symmetric matrix
• B. skew symmetric matrix
• C. diagonal matrix
• D. null matrix

Answer 1

Answer: (B)

skew symmetric matrix

We are given $A^{\prime }=A,B{}^{\prime }=B$
Now, $(AB-BA{)}^{\prime }=(AB{)}^{\prime }-(BA{)}^{\prime }$
$=B{}^{\prime }{A}^{\prime }-A^{\prime }B{}^{\prime }$
$=BA-AB$
$=-(AB-BA)$
i.e. $(AB-BA{)}^{\prime }=-(AB-BA)$
Hence, $AB-BA$ is a skew symmetric matrix.
Hence, option B is correct.