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Let $A and B$ be two $3 \times 3$ matrices of real number where $A$ is symmetric and $B$ is skew symmetric. If $(A + B) (A - B) = (A - B) (A + B)$ and ${(A B)}^{T} = {(- 1)}^{K} A B$ then the possible values of $K$ are

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answer is B.

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Detailed Solution

$A^{T} = A a n d B^{T} = - B (A + B) (A - B) = (A - B) (A + B) A^{2} - A B + B A - B^{2} = A^{2} + A B - B A - B^{2} A B = B A {(A B)}^{T} = {(- 1)}^{K} A B B^{T} A^{T} = {(- 1)}^{K} A B - B A = {(- 1)}^{K} A B - A B = {(- 1)}^{K} A B$

K is odd

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