Question

Let $A$ and $B$ be $3\times 3$ matrices such that $A^T=-A,B^T=B$, then matrix $(\lambda AB+3BA)$ is a skew symmertric matrix for

• A. $\lambda =3$
• B. $\lambda =-3$
• C. $\lambda =3$ or $\lambda =-3$
• D. $\lambda =3$ and $\lambda =-3$

Answer 1

The correct option is A $\lambda =3$

If the matrix $\lambda AB+3BA$ to be skew-symmetry,

Then, it must satisfy ${(\lambda AB+3BA)}^T=-(\lambda AB+3BA)$

Now, considering ${(\lambda AB+3BA)}^T$
$=\lambda {\left(AB\right)}^T+3{\left(BA\right)}^T$
$=\lambda B^T{A}^T+3A^T{B}^T$
Substituting values of $A^T=-A$ and $B^T=B$, we get
$=-\lambda BA-3AB=-(3AB+\lambda BA)$

From this, we get $\lambda =3$ clearly.

Hence, option A.