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If A=[[cos theta,sin theta],[-sin theta,...

If A=`[[cos theta,sin theta],[-sin theta,cos theta]], ` then `lim _(n rarr infty )A^(n)/n ` is (where `theta in R`)

A

a zero matrix

B

an identity matrix

C

`[[0,1],[-1,0]]`

D

`[[0,1],[0,-1]]`

Text Solution

Verified by Experts

The correct Answer is:
A
`because A = [[cos theta , sin theta],[-sin theta, cos theta ]]`
`therefore A^(n) = [[cos ntheta , sin ntheta],[-sin ntheta, cos ntheta ]]`
`rArr A^(n)/n = [[lim_(nrarr infty)(cos ntheta)/n , lim_(nrarr infty)(sin ntheta)/n],[-lim_(nrarr infty)(sin ntheta)/n, lim_(nrarr infty)(cos ntheta)/n ]]= [[0,0],[0,0]]`
= a zero matirx `[because - 1 lt sin infty 1 and -1 lt cos infty lt 1]`
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