Question

Let $A$ and $B$ are $3\times 3$ matrices with real number entries, where $A$ is symmetric, $B$ is skew-symmetric and $\left(A+B\right)\left(A-B\right)=\left(A-B\right)\left(A+B\right)$ . If ${\left(AB\right)}^T={\left(-1\right)}^{k}AB$ , then the sum of all possible integral value of $k$ in $\left[2,10\right]$ is equal to (where $A^T$ represent transpose of matrix $A$ )

• A. $20$
• B. $24$
• C. $28$
• D. $45$

Answer 1

Answer: (B)

$A^T=A,B^T=-B$
$\left(A+B\right)\left(A-B\right)=\left(A-B\right)\left(A+B\right)$
$\Rightarrow A^2-AB+BA-B^2=A^2+AB-BA-B^2$
$\Rightarrow AB=BA\dots \dots \left(i\right)$
Now, ${\left(AB\right)}^T=B^T{A}^T=-BA=-AB={\left(-1\right)}^{k}AB$
$\Rightarrow k$ is an odd number
$\Rightarrow k=3,5,7,9$
Sum $=24$