Reduced row echelon form with variables

Question

I'm new to this, but if I have the matrix \begin{equation} A= \begin{bmatrix}1&2&3&1\\2&1&1&x^2+x \\ 3&6&x&x-6\end{bmatrix}\end{equation} and if I want to use the reduced row echelon form formula, do I have to get the $x$ variables to $0$, or could they stay in the matrix?

Answer 1

You must be careful when row reducing with unknown parameters in the matrix...you can row reduce, but beware of errors such as division by zero, for example if you multiply the last row by $x^{-1}$ you must be sure that $x \neq 0$, else that row operation is undefined. You get the same problem if you multiply by $(x-6)^{-1}$ (and perhaps it is possible that $x=6$, in which case you have division by zero again), etc.

So my advice in this case: row reduce normally until you have row echelon form (without dividing by any factors containing $x$). You won't be able to go all the way to RRE.