Prove the orthogonal matrices of order two are of the form `[(cos theta,-sin theta),(sin theta,cos theta)]` or `[(cos theta,sin theta),(sin theta,-cos

Monjur Alahi

Prove the orthogonal matrices of order two are of the form `[(cos theta,-sin theta),(sin theta,cos theta)]` or `[(cos theta,sin theta),(sin theta,-cos theta)]`


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Salim Ahmed Kawsar

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Let matrix `A=[(a,b),(c,d)]` is orthogonal matrix.
`:. [(a,b),(c,d)][(a,c),(b,d)]=[(1,0),(0,1)]`
`implies [(a^(2)+b^(2),ac+bd),(ac+bd,c^(2)+d^(2))]=[(1,0),(0,1)]`
`implies a^(2)+b^(2)=1` (1)
`c^(2)+d^(2)=1` (2)
`ac+bd=0` (3)
`implies a/d= (-b)/c= k` (let)
`implies c^(2)+d^(2)=(a^(2)+b^(2))/k=1//k^(2)` or `k^(2)=1` or `k= pm 1`
`implies a/b=(-b)/c =pm 1`
Also, we must have `a, b, c, d in [-1, 1]` for equations (1) and (2) to get defined.
Hence, without loss of generality, we can assume `a= cos theta` and `b= sin theta`
So, for `a/d=(-b)/c=1`, we have `[(a,b),(c,d)] equiv [(cos theta,- sin theta),(sin theta,cos theta)]` and For `a/d=(-b)/c=-1`, we have `[(a,b),(c,d)] equiv [(cos theta,sin theta),(sin theta,-cos theta)]`

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