Prove that the product of the matrices `[[cos^2alpha, cosalphasinalpha],[cosalphasinalpha,sin^2alpha]]` and `[[cos^2beta,cosbetasinbeta],[cosbetasinbeta,sin^beta]]` is the null matrix when `alpha and beta` differ by an odd multiple of `pi/ 2`.

Options

A. 0
B. multiple of `pi`
C. an odd multiple of `pi//2`
D. None of these

Answer

C

Explanation

`AB=[{:(" "cos^(2)alpha,,cosalphasinalpha),(cosalphasinalpha,," "sin^(2)alpha):}]xx[{:(" "cos^(2)beta,,cosbetasinbeta),(cosbetasinbeta,," "sin^(2)beta):}]=[{:(cosalphacosbetacos(alpha-beta),,cosalphasinbetacos(alpha-beta)),(sinalphacosbetacos(alpha-beta),,sinalphasinbetacos(alpha-beta)):}]`
`=[{:(0,0),(0,0):}]`
`implies" "cos(alpha-beta)=0`
`implies" "alpha-beta=(2n+1)pi//2`