If-X-3-4-1-1-the-value-of-X-n-is-

Question Number 41141 by Kishan Daroga last updated on 02/Aug/18

If X = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}], the value of X^{n} is

Commented by math khazana by abdo last updated on 04/Aug/18

l e t A = (\begin{matrix} 3 - 4 \\ 1 - 1 \end{matrix}) t h e c a r a c t e r i s t i c p o l y n o m e o f

A i s p (x) = d e t (A - x I) = | \begin{matrix} 3 - x - 4 \\ 1 - 1 - x \end{matrix} |

= (x - 3) (x + 1) + 4 = x^{2} + x - 3 x - 3 + 4

= x^{2} - 2 x + 1 = {(x - 1)}^{2} s o 1 i s d o u b l e p r o p e r

v a l u e f o r A

v (1) = k e r (A - I) = {u / (A - I) u = 0}

l e t u (\begin{matrix} x \\ y \end{matrix}) \Rightarrow (\begin{matrix} 2 - 4 \\ 1 - 2 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0 \Rightarrow

{_{x - 2 y = 0}^{2 x - 4 y = 0} \Rightarrow x = 2 y \Rightarrow (x, y) = (2 y, y) = y (2, 1)

s o v (1) = D_{e} w i t h v e c t o r e (2, 1) i t s c l e a r t h a t

A i s n o t d i a g o n a l i s a b l e b y c a y l e y h a m i l t o n

t h e o r e m g i v e A^{2} - 2 A + I = 0 \Rightarrow

A^{2} = 2 A - I \Rightarrow A^{3} = (2 A - I) A = 2 A^{2} - A

= 2 (2 A - I) - A = 3 A - 2 I f o r t h a t w e m u s t

d e t e r m i n e t h e s e q u e n c e s u_{n} a n d v_{n} w i t h

v e r i f y A^{n} = u_{n} A + v_{n} I \Rightarrow

A^{n + 1} = u_{n + 1} A + v_{n + 1} I = (u_{n} A + v_{n} I) A

= u_{n} A^{2} + v_{n} A = u_{n} (2 A - I) + v_{n} A

= (2 u_{n} + v_{n}) A - u_{n} I \Rightarrow

u_{n + 1} = 2 u_{n} + v_{n} a n d v_{n + 1} = - u_{n} \dots b e c o n t i n u e d \dots