आव्यूह `A={:[(cos theta,sintheta,0),(-sin theta,cos theta ,0),(0,0,1)]:}` का प्रतिलोम क्या है ?

Options

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

दिया है,
`A={:[(cos theta,sin theta,0),(-sin theta,cos theta,0),(0,0,1)]:}`
`|A|=1[cos ^(2)theta-(-sin^(2)theta)]=1!=0`
`C_(11)=|(cos theta,0),(0,1)|=cos theta`
`C_(12)=-|(-sin theta,0),(0,1)|=sin theta`
`C_(13)=|(-sin theta , cos theta),(0,0)|=0`
`C_(21)=-|(sin theta,0),(0,1)|=-sin theta`
`C_(22)=|(cos theta,0),(0,1)|=cos theta `
`C_(23)=-|(cos theta, sin theta),(0,0)|=0`
`C_(31)=|(sin theta,0),(cos theta,0)|=0`
`C_(32)=-|(cos theta,0),(-sin , 0)|=0`
`C_(33)=|(cos theta, sin theta),(-sin theta, cos theta)|=cos^(2) theta+sin^(2) theta=1`
`:."adj"A=[(cos theta,sintheta,0),(-sin theta, cos theta,0),(0,0,1)]^(T)`
`=[(cos theta,-sin theta,0),(sin theta,cos theta,0),(0,0,1)]`
`:.A^(-1)=("adj" A)/(|A|)`
`=1/1[(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)]`