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 $\left(A B\right)^{T}=\left(- 1\right)^{k}AB$ , 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)

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