Soft Question: Limitations of matrix use in proof demonstration?

Question

It is common routine in algebra to make variable changes in proof demonstration. For instance, to examine the relationship between 2 variables, we often set $$x+y=u$$ $$x-y=v$$ For 3 variables, $$x+y=u$$ $$x+z=v$$ $$y+z=w$$ These linear transformations are obviously the scalar multiplication: $$(u,v)={{(x, y )} \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix}}$$ and $$(u, v,w )= {{(x, y,z )}\begin{bmatrix} 1&1&0\\ 1&0&1\\ 0&1&1 \end{bmatrix}}$$ My question is, besides the obvious restriction that the determinant of these matrices must be non-zero, are there any other ones? For instance,can one use matrices that are non-square as well?