If `A=[(cos^(2)alpha, cos alpha sin alpha),(cos alpha sin alpha, sin^(2)alpha)]` and `B=[(cos^(2)betas,cos beta sin beta),(cos beta sin beta, sin^(2) beta)]` are two matrices such that the product AB is null matrix, then `alpha-beta` is

Options

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

Answer

C

Explanation

`"Given"AB=0`
`therefore [{:(,cos^(@)alpha,cos alpha sin alpha),(,cos alpha sin alpha, sin^(2)alpha):}]` `xx [{:(,cos^(2)beta,cos beta sin beta),(,cos beta sin beta,sin^(2)beta):}]` `=[{:(,0,0),(,0,0):}]`
`Rightarrow [{:(,cos alpha cos beta cos(alpha-beta)),(,cos beta sin alpha cos (alpha-beta)):}` `{:(,cos alpha sin beta cos (alpha-beta)),(,sin alpha sin beta cos (alpha-beta)):}]` `=[{:(,0,0),(,0,0):}]`
`Rightarrow cos(alpha-beta)=0`
`Rightarrow alpha-beta` is an odd multiple of `pi//2`.