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Question
Answer the following question:
If A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]` and A + AT = I, where I is unit matrix 2 × 2, then find the value of α
Solution
A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]`
∴ AT = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`
∴ A + AT = I gives
`[(cosalpha, -sinalpha),(sinalpha, cosalpha)] + [(cosalpha, sinalpha),(-sinalpha, cosalpha)] = [(1, 0),(0, 1)]`
∴ `[(2cosalpha, 0),(0, 2cosalpha)] = [(1, 0),(0, 1)]`
∴ by equality of matrices,
2cos α = 1
∴ cos α = `1/2`
∴ cos α = `cos pi/3`
∴ α = `pi/3 or 60^circ`
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