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...

 

[Office_Shredder]

What about the a?

 

[cookiesyum]

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)?