Question

Consider the matrix A_α = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Find all possible real values of α satisfying the condition `"A"_alpha + "A"_alpha^"T"` = I

Answer 1

A_α = `[(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"`