(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Compute A^{17}.

1 0 4

A=0 1 2

0 0 4

2. Relevant equations

A=PDP^{-1}(D is a diagonal matrix)

3. The attempt at a solution

I got eigenvalues 1 and 4, and corresponding eigenspaces u_{1}=[1,0,0]^{T}, u_{2}=[0,1,1]^{T}and u_{3}=[4/3, 2/3 ,1]^{T}.

So, I computed P= (1/3) 3 0 4

0 3 2

0 0 3

also, P^{-1}= (1/3) 3 0 -4

0 3 -2

0 0 3.

D= 1/3 1 0 0

0 1 0

0 0 4.

and then A^{17}= PD^{17}P^{-1}

= 1/9 3 0 4[itex]\cdot[/itex]4^{17}3 0 -4

0 3 2[itex]\cdot[/itex]4^{17}0 3 -2

0 0 3[itex]\cdot[/itex]4^{17}0 0 3

= 1 0 4/3(4^{17}-1)

0 1 2/3(4^{17}-1)

0 0 4^{17}

sorry for the matrix form here. I couldn't find any commands to do it correctly.

Am I doing right here?

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# Homework Help: Linear algebra- diagonalizability

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