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Question
Consider the matrix Aα = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Find all possible real values of α satisfying the condition `"A"_alpha + "A"_alpha^"T"` = I
Solution
Aα = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]`
`"A"_alpha^"T" = [(cos alpha, sin alpha),(-sin alpha, cos alpha)]`
`"A"_alpha + "A"_alpha^"T" = [(cos alpha, - sin alpha),(sin alpha, cos alpha)] + [(cos alpha, sin alpha),(-sin alpha, cos alpha)]`
= `[(cos alpha + cos alpha, -sinalpha + sin alpha),(sin alpha - sin alpha, cos alpha + cos alpha)]`
`"A"_alpha + "A"_alpha^"T" = [(2cos alpha, 0),(0, 2 cos alpha)]`
Given `"A"_alpha + "A"_alpha^"T"` = I
∴ `[(2cos alpha, 0),(0, 2 cos alpha)] = [(1, 0),(0, 1)]`
`2 cos alpha` = 1
⇒ `cos allpha = 1/2`
The general solution is α = `2"n" pi +- pi/3, "n" ∈ "Z"`
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