Consider the following systems along with the corresponding substitutions:
\begin{align} \text{System 1} \\ 2x^2 + 3y + 5z^z & = 7 \\ 4x^2 + 9y + 10z^z & = 2 \\ x^2 + y = 2z^z & = 1 \end{align} Suppose we let $u = x^2$ and $v = z^z$, yielding 3 equations with 3 outside variables.
\begin{align} \text{System 2} \\ 4x - 3xy + 7y = 5 \\ 8x + 11xy - 9y = 3 \end{align} Suppose we let $u = xy$, yielding 2 equations with 3 outside variables.
\begin{align} \text{System 3} \\ -2\sin(x) + 6y = 1 \\ 3x + 3y = 4 \\ \sin(x) + 2y = -12 \end{align} Suppose we let $u = sin(x)$, yielding 3 equations with 3 outside variables.
The non-linearity in each system is masked by applying the substitution, so we could then convert each system into a matrix. We can make similar substitutions prior to differentiating, in which case the chain rule is used to take account of the fact that $u$ is a dependent variable, dependent upon the variable of differentiation as defined by the substitution. The chain rule ensures that the information in the outside expression and the information in the inside expression (the substitution definition) are both taken into account during differentiation. Is there an analogous rule that can be used to perform elementary row operations on matrices with substituted/dependent variable coefficients?