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]]`

Options

A. a zero matrix
B. an identity matrix
C. `[[0,1],[-1,0]]`
D. `[[0,1],[0,-1]]`

Answer

A

Explanation

`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]`