Similar Matrices Minimal Polynomial of A^3

[talolard]

Homework Statement

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

Homework Equations

The Attempt at a Solution

A is similar to a diagnol matrix with 1,2,3,4 on the diagnol, let's 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 $$A=P^{-1}BP$$
$$A^2=P^{-1}BPP^{-1}BP=P^{-1}B^2P$$
$$A^3=P^{-1}B^2PP^{-1}BP=P^{-1}B^3P$$

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?

 

[lanedance]

that sounds reasonable to me, to see it another way, assume u is an eiegnevector of A with e-val lambda, and consider the action of A^3
$$A^3u = AAAu = AA\lambda u= A\lambda^2 u= \lambda^3 u$$

so lambda^3 is an eigenvalue of A^3, wth eignevector u