Let `A and B` be `3xx3` matrtices of real numbers, where `A` is symmetric, `"B"` is skew-symmetric , and `(A+B)(A-B)=(A-B)(A+B)dot` If `(A B)^t=(-1)^k A B , where . (A B)^t` is the transpose of the mattix `A B ,` then find the possible values of `kdot`

Options

A. 0
B. 1
C. 2
D. 3

Answer

B, D

Explanation

`because A^(t) = A, B^(t) = -B`
Given, `(A+B) (A-B) = (A-B) (A+B) `
`rArrA^(2) - AB + BA-B^(2) = A^(2) + AB - BA-B^(2)`
`rArr AB= BA`
Also, given `(AB)^(t)=(-1)^(k)AB`
`rArr B^(t) A^(t) = (-1)^(k) AB`
`rArr -BA = (-1)^(k) AB`
`rArr (-1) = (-1)^(k) [because AB= BA]`
`therefore k = 1, 3, 5, ...`