Row reduction to upper-triangular form over $F[x]$

The assertion is \(\boxed{\text{true}}\).

We prove it by induction on \(n\). The case \(n=1\) is immediate. Let \(A\in M_n(F[x])\). Since \(F[x]\) is a Euclidean domain, elementary row operations can replace the entries in the first column by their greatest common divisor \(d\) in the first position and zeros below it: repeated Euclidean division on pairs of entries gives exactly the usual gcd algorithm, implemented by invertible elementary row operations [@HK71]. Thus \(A\) is row-equivalent to \[ \begin{pmatrix} d&*\\ 0&B \end{pmatrix}, \] where \(B\) is an \((n-1)\times(n-1)\) matrix over \(F[x]\). By the induction hypothesis there is an invertible product of elementary row operations on the last \(n-1\) rows taking \(B\) to an upper-triangular matrix \(U\). Applying those same operations to the full matrix leaves the zeros in the first column below \(d\) unchanged and gives \[ \begin{pmatrix} d&*\\ 0&U \end{pmatrix}, \] which is upper triangular. Hence every matrix over \(F[x]\) is row-equivalent to an upper-triangular one.