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Question Number 91268 by mathmax by abdo last updated on 29/Apr/20

l e t A = (\begin{matrix} 1 2 \\ 1 - 1 \end{matrix})

1) c a l c u l s t e A^{n}

2) d e t e r m i n e e^{A} a n d e^{- 2 A}

3) f i n d c o s (A) a n d s i n A i s {c o s}^{2} A + {s i n}^{2} A = I ?

4) d e t e r m i n e s h (A) a n d c h (A) i s {c h}^{2} A - {s h}^{2} A = I ?

Commented by mathmax by abdo last updated on 29/Apr/20

1) t h e c a r a c t e r i s t i c p o l y n o m i s p (x) = d e t (A - x I)

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

= - 1 + x^{2} - 2 = x^{2} - 3 = (x - \sqrt{3}) (x + \sqrt{3})

p (x) = 0 \Leftrightarrow x = \bar{+} \sqrt{3}

x^{n} = Q p (x) + u_{n} x + v_{n} \Rightarrow {(\sqrt{3})}^{n} = u_{n} \sqrt{3} + v_{n}

{(- \sqrt{3})}^{n} = - \sqrt{3} u_{n} + v_{n} \Rightarrow {(\sqrt{3})}^{n} - {(- \sqrt{3})}^{n} = 2 \sqrt{3} u_{n} \Rightarrow

u_{n} = \frac{{(\sqrt{3})}^{n} - {(- \sqrt{3})}^{n}}{2 \sqrt{3}} a n d {(\sqrt{3})}^{n} + {(- \sqrt{3})}^{n} = 2 v_{n} \Rightarrow

v_{n} = \frac{{(\sqrt{3})}^{n} + {(- \sqrt{3})}^{n}}{2}

A^{n} = Q (A) p (A) + u_{n} A + v_{n} I = u_{n} A + v_{n} I (P (A) = 0 b y c a y l e y h a m i t o n)

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

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

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

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

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

Commented by mathmax by abdo last updated on 29/Apr/20

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

= (\begin{matrix} 3^{p} 0 \\ 0 3^{p} \end{matrix})

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

= (\begin{matrix} 3^{p} 2 . 3^{p} \\ 2 . 3^{p} - 3^{p} \end{matrix})

Commented by mathmax by abdo last updated on 29/Apr/20

s o r r y A^{2 p + 1} = (\begin{matrix} 3^{p} 2 . 3^{p} \\ 3^{p} - 3^{p} \end{matrix})

Commented by mathmax by abdo last updated on 29/Apr/20

2) e^{A} = \sum_{n = 0}^{\infty} \frac{A^{n}}{n!} = \sum_{n = 0}^{\infty} \frac{A^{2 n}}{(2 n)!} + \sum_{n = 0}^{\infty} \frac{A^{2 n + 1}}{(2 n + 1)!}

= \sum_{n = 0}^{\infty} \frac{1}{(2 n)!} (\begin{matrix} 3^{n} 0 \\ 0 3^{n} \end{matrix}) + \sum_{n = 0}^{\infty} \frac{1}{(2 n + 1)!} (\begin{matrix} 3^{n} 23^{n} \\ 3^{n} - 3^{n} \end{matrix})

= (\begin{matrix} \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n)!} 0 \\ 0 \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n)!} \end{matrix}) + (\begin{matrix} \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n + 1)!} 2 \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n + 1)!} \\ \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n + 1)!} - \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n + 1)!} \end{matrix})

Commented by mathmax by abdo last updated on 29/Apr/20

w e h a v e \sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n)!} = \sum_{n = 0}^{\infty} \frac{{(\sqrt{3})}^{2 n}}{(2 n)!} = c o s (\sqrt{3})

\sum_{n = 0}^{\infty} \frac{3^{n}}{(2 n + 1)!} = \frac{1}{\sqrt{3}} \sum_{n = 0}^{\infty} \frac{{(\sqrt{3})}^{2 n + 1}}{(2 n + 1)!} = \frac{1}{\sqrt{3}} s i n (\sqrt{3}) \Rightarrow

e^{A} = (\begin{matrix} c o s (\sqrt{3}) 0 \\ 0 c o s (\sqrt{3}) \end{matrix}) + (\begin{matrix} \frac{1}{\sqrt{3}} s i n (\sqrt{3}) \frac{2}{\sqrt{3}} s i n (\sqrt{3}) \\ \frac{1}{\sqrt{3}} s i n (\sqrt{3}) - \frac{1}{\sqrt{3}} s i n (\sqrt{3}) \end{matrix})

= (\begin{matrix} c o s (\sqrt{3}) + \frac{1}{\sqrt{3}} s i n (\sqrt{3}) \frac{2}{\sqrt{3}} s i n (\sqrt{3}) \\ \frac{1}{\sqrt{3}} s i n (\sqrt{3}) c o s (\sqrt{3}) - \frac{1}{\sqrt{3}} s i n (\sqrt{3}) \end{matrix})