यदि `A_(alpha) =[(cos alpha" "-sin alpha),(sinalpha" "cos alpha)]` तथा `A_(beta) =[(cosbeta" "-sinbeta),(sinbeta" "cosbeta)]` हो, तब निम्नलिखित में से कौन- सा एक सही है ?

Options

A. `A_((-alpha))*A_((-beta))=A_((alpha+beta))`
B. `A_((-alpha))*A_((beta))=A_((beta-alpha))`
C. `A_((alpha))+A_((-beta))=A_({-(beta-alpha)})`
D. `A_((alpha))+A_((beta)) = A_((alpha+beta))`

Answer

B

Explanation

`:' A_((alpha)) = [(cos alpha" "-sinalpha),(sin alpha" "cosalpha)]"तथा " A_((beta)) =[(cos beta" "sinbeta),(sin beta" "cosbeta)]`
` :. A_((-alpha))=[(cos alpha" "sin alpha),(-sin alpha" "cos alpha)]"तथा " A_((-beta))=[(cos beta" "sinbeta),(-sinbeta" "cosbeta)]`
`A_((-alpha))A_((-beta)) =[(cos alpha" "sinalpha),(-sinalpha" "cosalpha)][(cos beta" "sinbeta),(-sinbeta" "cosbeta)]`
`=[(cosalpha cos beta- sin alpha sin beta cos alpha sin beta+ sin alpha cos beta),(-sin alpha cos beta - cos alpha sin beta - sin alphasin beta + cos alpha cos beta)]`
` = [(cos(alpha+beta)" "sin(alpha+beta)),(-sin(alpha+beta)" "cos(alpha+beta))] = A_({-(alpha+beta)})`
`A_((-alpha))A_((beta)) = [(cos alpha" "sin alpha),(-sinalpha" "cos alpha)][(cosbeta" "-sinbeta),(sin beta" "cosbeta)]`
` =[(cos(beta-alpha)" "-sin(beta-alpha)),(n(beta-alpha)" "cos(beta-alpha))] = A_((beta-alpha))`