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?
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....