Finding the minimal polynomial of a matrix?

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Homework Statement



Let f(x) be an irreducible polynomial cubic in Q. For example

f(x) = ax^3 + bx^2 + cx + d

Let A be a 3 x 3 matrix with entries in Q such that char(A,x) = f(x). Find the minimal polynomial m(x) of A. Can you generalize to a degree n polynomial?


Homework Equations





The Attempt at a Solution



If the char(A,x) = f(x) then the companion matrix is...

[0, 0, -d]
[1, 0, -c]
[0, 1, -b]

Since the companion matrix's characteristic polynomial = its minimal polynomial, does this mean the minimal polynomial is just f(x). I'm missing something, aren't I...
 
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What about the a?
 
Office_Shredder said:
What about the a?

Oh. The companion matrix would be...

[0 0 -d/a]
[1 0 -c/a]
[0 1 -b/a]

so m(x) = x^3 + (b/a)x^2 + (c/a)x + (d/a)?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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