Question

If $A,B$ are symmetric matrices of same order then $AB-BA$ is a

• A. Symmetric matrix
• B. Skew-symmetric matrix
• C. Null matrix
• D. Unit matrix

Answer 1

Answer: (B)

Skew-symmetric matrix

Given $A$ and $B$ are symmetric

$A$ and $B$ are of same order

$AB$ and $BA$ exist

$A=A^T⟶1$

$B=B^T⟶2$

Now,

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

$=B^T{A}^T-A^T{B}^T$ (From $1$)

$\Rightarrow BA-AB$

$=-(AB-BA)$

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

$AB-BA$ is skew-symmetric matrix