Is every matrix in $F[x]^{n\times n}$ is row-equivalent to an upper-triangular matrix? I'm trying to solve this problem, but I don't have any idea of what I should do..


  True or false? Every matrix in $F[x]^{n\times n}$ is row-equivalent to an upper-triangular matrix.


This is a question in the section 7.4 of the book of linear algebra of Hoffman-Kunze.
See carefully the proof of Theorem 6. Do the same thing, then we get an upper triangular matrix which is row-equivalent to the original matrix.
(Only different thing is that we do not assume (ii), so the entries of the main diagonal may not be 1.)
I think I have an counter example; we know that $M_{3 \times 3}(\mathbb{R})\subseteq F[x]^{3 \times 3}$. The matrix:

$$ \left[
  \begin{array}{ c c c}
     0 &amp; 0 &amp; -1 \\
     1 &amp; 0 &amp; -3 \\
     0 &amp; 1 &amp; -8
  \end{array} \right]
$$

Because it's minimal polynomial is $x^3+8x^2+3x+1$, whose companion matrix is not row-equivalent to an upper-triangular matrix. is this right?