let-A-1-1-0-1-1-1-0-1-1-1-find-the-caractetistic-polynom-of-A-2-calculate-A-n-

Question Number 34305 by abdo mathsup 649 cc last updated on 03/May/18

l e t] A = (\begin{matrix} 1 1 0 \\ 1 1 1 \end{matrix})

(0 1 1

1) f i n d t h e c a r a c t e t i s t i c p o l y n o m o f A

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

Commented by prof Abdo imad last updated on 06/May/18

p_{c} (A) = d e t (A - X I) = | \begin{matrix} 1 - X 1 0 \\ 1 1 - X 1 \end{matrix} |

∣ 0 1 - X 1 - X ∣∣

= (1 - X) ({(1 - X)}^{2} - 1 + X) - 1 (1 - X) + 0

= (1 - X) (X^{2} - 2 X + 1 - 1 + X) - 1 + X

= (1 - X) (X^{2} - X) + X - 1

= - {(X - 1)}^{2} X + X - 1

= (X - 1) (1 - X (X - 1))

= (X - 1) (- X^{2} + X + 1) = (1 - X) (X^{2} - X - 1)

p_{c} (A) = 0 \Leftrightarrow X = 1 o r X^{2} - X - 1 = 0

Δ = 1 - 4 (- 1) = 5 \Rightarrow X_{1} = \frac{1 + \sqrt{5}}{2}

X_{2} = \frac{1 - \sqrt{5}}{2} . t h e p r o p e r v a l u e a r e λ_{1} = 1

λ_{2} = \frac{1 + \sqrt{5}}{2} a n d λ_{3} = \frac{1 - \sqrt{5}}{2}

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

b e c o n t i n u e d \dots . .