Let-A-be3-3-matrix-with-eigen-values-1-1-0-Then-determinant-of-I-A-100-

Question Number 59295 by pooja24 last updated on 07/May/19

L e t A b e 3 \times 3 m a t r i x w i t h e i g e n v a l u e s

1, - 1, 0 . T h e n d e t e r m i n a n t o f I + A^{100} = ? ?

Answered by alex041103 last updated on 10/May/19

b a s i c a l l y

A = {P D P}^{- 1}

w h e r e D i s a d i a g o n a l m a t r i x

\Rightarrow A^{100} = {P D}^{100} P^{- 1}

a l s o I = {P I P}^{- 1}

\Rightarrow M = I + A^{100} = P (I + D^{100}) P^{- 1}

T h e n d e t (M) = d e t (P) d e t (I + D^{100}) d e t (P^{- 1}) =

= [d e t (P) d e t (P^{- 1})] d e t (I + D^{100}) =

= d e t ({P P}^{- 1}) d e t (I + D^{100}) =

= d e t (I) d e t (I + D^{100}) =

= d e t (I + D^{100}) = d e t (M)

W e d e f i n e P a s t h e m a t r i x w h i c h g e t s

u s f r o m t h e n o r m a l s y s t e m o f c o o r d i n a t e s t o t h e

s y s t e m o f t h e c o o r d i n a t e s w h i c h u s e s

t h e e i g e n v e c t o r s a s b a s i s v e c t o r s . T h e n

D i s a d i a g o n a l m a t r i x w i t h t h e e i g e n

v a l u e s i n i t .

\Rightarrow D = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]

\Rightarrow D^{100} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

\Rightarrow d e t (M) = | \begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} | = 4

\Rightarrow d e t (I + A^{100}) = 4