If `A=[cosalphasinalpha-sinalphacosalpha]` , then verify that `A^T\ A=I_2` .

Options

A. Null matrix
B. Identify matrix
C. A
D. `-A`

Answer

B

Explanation

`A=[{:(cos alpha,sin alpha),(-sin alpha,cos alpha):}]`
`"AA"^(T)=[{:(cos alpha,sin alpha),(-sin alpha,cos alpha):}][{:(cos alpha,-sin alpha),(sin alpha,cos alpha):}]`
`=[{:(" "cos^(2)alpha+sin^(2)alpha,-cos alpha sin alpha+sin alpha cos alpha),(-sin alpha cos alpha+sin alpha cos alpha," "sin^(2)alpha+cos^(2)alpha):}]`
`=[{:(1,0),(0,1):}]`