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Question
If A = `[(cos alpha, sin alpha), (-sin alpha, cos alpha)]` then verify that A' A = I
Solution
Given, A = `[(cos alpha, sin alpha), (-sin alpha, cos alpha)]`
So, A' =`[(cos alpha, -sin alpha), (sin alpha, cos alpha)]`
Now, A' A = `[(cos alpha, -sin alpha), (sin alpha, cos alpha)] xx [(cos alpha, sin alpha), (-sin alpha, cos alpha)]`
`= [(cos^2 alpha+ sin^2 alpha, cos alpha sin alpha - sin alpha cos alpha),(sin alpha cos alpha - cos alpha sin alpha, sin^2 + cos^2 alpha)]`
`= [(1,0),(0,1)] = I ...["Because" sin^2 alpha + cos^2 alpha = 1]`
Hence, A' A = I
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