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Matrices

Question
CBSEENMA12034299
Wired Faculty App

If space straight A subscript straight alpha space end subscript equals space open square brackets table row cell cosα space end cell cell space space sin space straight alpha space end cell row cell negative sin space straight alpha space end cell cell space space cos space straight alpha space end cell end table close square brackets comma space then space show space that straight A subscript straight alpha space end subscript. space straight A subscript straight beta space end subscript equals straight A subscript straight alpha space end subscript plus space straight A subscript straight beta space end subscript space and space left parenthesis straight A subscript straight alpha space end subscript right parenthesis to the power of straight n space space equals straight A subscript n space alpha end subscript space where space straight n space element of space straight N.

Solution
Here space space straight A subscript straight alpha space end subscript equals space open square brackets table row cell cos space straight alpha space end cell cell space space sin space straight alpha space end cell row cell negative sin space straight alpha space end cell cell space space cos space straight alpha space end cell end table close square brackets space comma space straight A subscript straight beta space space end subscript equals space space open square brackets table row cell cos space straight beta space space end cell cell space space sin space straight beta space space end cell row cell negative sin space straight beta space end cell cell space space cos space straight beta space end cell end table close square brackets therefore space space straight A subscript straight alpha space end subscript. space straight A subscript straight beta space space end subscript equals space space open square brackets table row cell cos space straight alpha space end cell cell space space sin space straight alpha space end cell row cell negative sin space straight alpha space end cell cell space space cos space straight alpha space end cell end table close square brackets space space open square brackets table row cell cos space straight beta space space end cell cell space space sin space straight beta space space end cell row cell negative sin space straight beta space end cell cell space space cos space straight beta space end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space equals space open square brackets table row cell cos space straight alpha space cos space space straight beta minus space sin space straight alpha space sin space straight beta end cell cell space space cos space straight alpha space sin space straight beta space space plus sin space straight alpha space cos space space straight beta space space end cell row cell negative sin space straight alpha space cos space space straight beta space minus cos space straight alpha space sin space straight beta space end cell cell negative sin space straight alpha space sin space straight beta space plus cos space straight alpha space space cos space space straight beta space space space end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space equals space space open square brackets table row cell cos space left parenthesis straight alpha plus space straight beta space right parenthesis space end cell cell space space space sin space straight beta space left parenthesis straight alpha plus space straight beta space right parenthesis space end cell row cell negative sin space straight beta space left parenthesis straight alpha plus space straight beta space right parenthesis end cell cell space space cos space straight beta space left parenthesis straight alpha plus space straight beta space right parenthesis end cell end table close square brackets space equals space straight A subscript straight alpha plus straight beta end subscript therefore straight A subscript alpha. straight A subscript beta space end subscript equals straight A subscript alpha plus space straight beta space Similarly space straight A subscript alpha. space straight A subscript beta. space Aγ space equals straight A subscript alpha plus space subscript straight beta plus subscript straight gamma space end subscript space Proceeding space in space this space way comma space we space get comma space straight A semicolon subscript straight alpha space. space straight A space subscript straight beta space end subscript. straight A subscript gamma space end subscript... space to space straight n space factors space equals straight A subscript alpha space plus space subscript straight beta space end subscript plus subscript space straight gamma end subscript space plus space... space plus space to space straight n space terms space Put space straight alpha space equals space straight beta space equals space straight gamma space equals... space therefore space straight A space subscript straight alpha space straight A subscript space straight alpha end subscript space straight A space subscript straight alpha space end subscript comma..... to space factors space equals straight A subscript alpha space end subscript plus subscript space straight alpha space end subscript plus space subscript straight alpha space end subscript plus space plus space to space straight n space terms space space or space left parenthesis straight A subscript alpha right parenthesis straight n space equals space straight A subscript n space alpha end subscript where space straight n space element of space straight I.

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