Question

prove its a orthogonal matrix
cos theta cos theta
sin theta cos theta

Answer 1

Let matrix A=[acbd] is orthogonal matrix.

∴[acbd][abcd]=[1001]

⇒[a2+b2ac+bdac+bdc2+d2]=[1001]

⇒a2+b2=1 (1)

c2+d2=1 (2)

ac+bd=0 (3)

⇒ad=−bc=k (let)

⇒c2+d2=a2+b2k=1/k2 or k2=1 or k=±1

⇒ab=−bc=±1

Also, we must have a,b,c,d∈[−1,1] for equations (1) and (2) to get defined.

Hence, without loss of generality, we can assume a=cosθ and b=sinθ

So, for ad=−bc=1, we have [acbd]≡[cosθsinθ−sinθcosθ] and For ad=−bc=−1, we have