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If A= [a(ij)] (nxxn) and f is a functi...

If `A= [a_(ij)] _(nxxn)` and f is a function, we define `f (A) = [f(a_(ij))] _(nxxn)."Let"A = [[pi/2-theta,theta],[-theta,pi/2-theta]]` then

A

sin A is invertible

B

sin A = cos A

C

sin A is orthogonal

D

sin 2 A=2 sin A cos A

Text Solution

Verified by Experts

The correct Answer is:
A, C
`sin A = [[cos theta ,sin theta],[-sin theta , cos theta]] and cos theta = [[sin theta , cos theta ],[cos theta , sin theta ]]`
`therefore abs(sin A) cos^(2) theta + sin ^(2) theta = 1 ne 0 `
Hence, sin A is incertible.
Also, `(sinA)(sin A)^(T) = [[cos theta ,sin theta],[-sin theta , cos theta]][[cos theta ,-sin theta],[sin theta , cos theta]]`
` = [[1,0],[0,1]] = I`
Hence, sin A is orthogonal.
Also ,`2sinA" " sin A = 2[[cos theta ,sin theta],[-sin theta , cos theta]][[sin theta,cos theta ],[ cos theta,sin theta]]`
`= 2 [[sin 2theta, 1 ],[cos 2 theta, 0]] ne sin 2 A`
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