If the matrix `[{:(cos theta,sin theta,0),(sin theta,cos theta,0),(0,0,1):}]` is singular, then what is one of the values of `theta`?

Options

A. `({:(cos theta,-sin theta,0),(sin theta,cos theta,0),(" "0," "0,1):})`
B. `({:(cos theta,sin theta,-sin theta),(" "0," "1," "0),(sin theta," "0,cos theta):})`
C. `({:(1," "0," "0),(0,cos theta,-sin theta),(0,sin theta,cos theta):})`
D. `({:(cos theta,sin theta,0),(-sin theta,cos theta,0),(" "0," "0,1):})`

Answer

A

Explanation

`A=[{:(cos theta,sin theta,0),(-sin theta,cos theta,0),(" "0," "0,1):}]`
We know, `"AA"^(-1) = I`
Let us take first option (a) as `bar(A)^(1)`.
`[{:(cos theta,sin theta,0),(-sin theta,cos theta,0),(" "0," "0,1):}][{:(cos theta,-sin theta,0),(sin theta,cos theta,0),(" "0," "0,1):}]`
`[{:(" "cos^(2)theta+sin^(2)theta,-sin theta cos theta+cos theta sin theta,0),(-sin theta cos theta + cos theta sin theta," "sin^(2)theta+cos^(2)theta,0),(" "0," "0,1):}]`
`=[{:(1,0,0),(0,1,0),(0,0,1):}]=I`