Γ(θ)= [((cos θ),(sin θ)),((−sin θ),(cos θ)) ] Λ(θ,t)= [((cos θ),(sinh t sin θ)),((sin θ),(cosh t cos θ)) ] ζ(θ,t)=Γ(θ)×Λ(θ,t)+Λ(θ,t)×Γ(θ) ζ(θ,0)=? det ζ(θ,0)=?


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Question Number 346 by 123456 last updated on 25/Jan/15

$Γ (θ) = [\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}]$ $Λ (θ, t) = [\begin{matrix} \cos θ & \sinh t \sin θ \\ \sin θ & \cosh t \cos θ \end{matrix}]$ $ζ (θ, t) = Γ (θ) \times Λ (θ, t) + Λ (θ, t) \times Γ (θ)$ $ζ (θ, 0) = ?$ $\det ζ (θ, 0) = ?$
	Answered by prakash jain last updated on 23/Dec/14

	$Λ (θ, 0) = [\begin{matrix} \cos θ & 0 \\ \sin θ & \cos θ \end{matrix}]$ $Γ (θ) \times Λ (θ, 0) = [\begin{matrix} \cos^{2} θ + \sin^{2} θ & 0 + \cos θ \sin θ \\ - \sin θ \cos θ + \sin θ \cos θ & \cos^{2} θ \end{matrix}]$ $= [\begin{matrix} 1 & \cos θ \sin θ \\ 0 & \cos^{2} θ \end{matrix}]$ $Λ (θ, 0) \times Γ (θ) = [\begin{matrix} \cos^{2} θ + 0 & \cos θ \sin θ + 0 \\ \cos θ \sin θ - \sin θ \cos θ & 1 \end{matrix}]$ $= [\begin{matrix} \cos^{2} θ & \cos θ \sin θ \\ 0 & 1 \end{matrix}]$ $Γ (θ) \times Λ (θ, 0) + Λ (θ, 0) \times Γ (θ) = [\begin{matrix} 1 + \cos^{2} θ & 2 \cos θ \sin θ \\ 0 & 1 + \cos^{2} θ \end{matrix}]$ $ζ (θ, 0) = [\begin{matrix} 1 + \cos^{2} θ & 2 \cos θ \sin θ \\ 0 & 1 + \cos^{2} θ \end{matrix}]$ $\det ζ (θ, 0) = {(1 + \cos^{2} θ)}^{2}$
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