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

Let F(x)=(x-1)(x-2)(x-3)(x-4) be the charecteristic polynomial of A. Find the minimal polynomial of A^3

2. Relevant equations

3. The attempt at a solution

A is similar to a diagnol matrix with 1,2,3,4 on the diagnol, lets call it B.

We know that A is diagnizable because The degree of the charecteristic polynomial of a matrix is always equal to its size, and if a matrix of size n has n eigenvalues then it si diagnizable

So [tex] A=P^{-1}BP [/tex]

[tex] A^2=P^{-1}BPP^{-1}BP=P^{-1}B^2P [/tex]

[tex] A^3=P^{-1}B^2PP^{-1}BP=P^{-1}B^3P [/tex]

And so B^3 is a diagnizable amtrix with 1, 8, 27, 64 on its diagnol, which means that its charecteristic polynomial is g(x)=(1-x)(8-x)(27-x)(64-x) and because each value in the charecteristic polynomial must apear in the minimal polynomial g(x) si also the minimal polynomial of A^3.

Is that correct?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Similar matrices

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