If `{:E(theta)=[(cos^2 theta,costhetasintheta),(costhetasintheta,sin^2theta)]:},and thetaand phi` differ by an odd multiple of `pi//2," then "E(theta)E(phi)` is a

Options

A. unit matrix
B. null matrix
C. diagonal matrix
D. None of these

Answer

B

Explanation

`E(theta)E(phi)=[(cos^(2)theta, cos theta sin theta),(cos theta sin theta,sin^(2)theta)]`
`xx[(cos^(2) phi, cos phi sin phi),(cos phi sin phi, sin^(2)phi)]`
`=[(cos^(2) theta cos^(2)phi+cos theta sin theta cos phi sin phi, cos^(2) theta cos phi si phi+cos theta sin theta sin^(2)phi),(-cos theta sin theta cos^(2)phi+sin^(2)theta cos phi sin phi, cos theta sin theta cos phi sin phi + sin^(2) theta sin^(2)phi)]`
`=[(cos theta cos phi cos (theta-phi), cos theta sin phi cos (theta -phi)),(cos phi sin theta cos (theta -phi),sin theta sin phi cos (theta -phi))]`
`[ :' cos (A-B)=cos A cos B+sin A sin B]`
`=[(cos theta cos phi cos (2n+1)(pi)/2, cos theta sin phi cos (2n+1)(pi)/2),(cos phi sin theta cos (2n+1)(pi)/2, sin theta sin phi cos (2n+1)(pi)/2)]`
`=[(0,0),(0,0)] [ :' cos (2n+1)(pi)/2=0]`