let-A-2-1-1-3-1-calculate-A-n-2-find-e-A-e-A-3-determine-ch-A-and-sh-A-is-ch-2-A-sh-2-A-1-

Question Number 100994 by mathmax by abdo last updated on 29/Jun/20

let A = (\begin{matrix} 2 1 \\ 1 3 \end{matrix})

1) calculate A^{n}

2) find e^{A}, e^{- A}

3) determine ch (A) and sh (A) is {ch}^{2} A - {sh}^{2} A = 1 ?

Answered by mathmax by abdo last updated on 01/Jul/20

1) the caracteristic polynom of A is p_{c} (x) = \det (A - xI)

= | \begin{matrix} 2 - x 1 \\ 1 3 - x \end{matrix} | = (x - 2) (x - 3) - 1 = x^{2} - 5 x + 6 - 1 = x^{2} - 5 x + 5

P_{c} (x) = 0 \Leftrightarrow x^{2} - 5 x + 5 = 0 \to Δ = 25 - 20 = 5 \Rightarrow α = \frac{5 + \sqrt{5}}{2} and β = \frac{5 - \sqrt{5}}{2}

we have x^{n} = {qP}_{c} (x) + u_{n} x + v_{n} \Rightarrow {\begin{cases} α^{n} = u_{n} α + v_{n} \\ β^{n} = u_{n} β + v_{n} \end{cases}

\Rightarrow u_{n} = \frac{α^{n} - β^{n}}{α - β} = \frac{1}{\sqrt{5}} {{(\frac{5 + \sqrt{5}}{2})}^{n} - {(\frac{5 - \sqrt{5}}{2})}^{n}}

v_{n} = α^{n} - α u_{n} = α^{n} - α \times \frac{α^{n} - β^{n}}{α - β} = \frac{α^{n + 1} - β α^{n} - α^{n + 1} + α β^{n}}{α - β} = \frac{α β^{n} - β α^{n}}{α - β}

= \frac{\frac{5 + \sqrt{5}}{2} {(\frac{5 - \sqrt{5}}{2})}^{n} - \frac{5 - \sqrt{5}}{2} {(\frac{5 + \sqrt{5}}{2})}^{n}}{\sqrt{5}}

we have A^{n} = u_{n} A + v_{n} I (by cayley hamilton) \Rightarrow

A^{n} = \frac{1}{\sqrt{5}} {{(\frac{5 + \sqrt{5}}{2})}^{n} - {(\frac{5 - \sqrt{5}}{2})}^{n}} (\begin{matrix} 2 1 \\ 1 3 \end{matrix}) + \frac{\frac{5 + \sqrt{5}}{2} {(\frac{5 - \sqrt{5}}{2})}^{n} - \frac{5 - \sqrt{5}}{2} {(\frac{5 + \sqrt{5}}{2})}^{n}}{\sqrt{5}} (\begin{matrix} 1 0 \\ 0 1 \end{matrix})