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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....