Adding two rows in an augmented matrix: does this always preserve the set of solutions? Given a system of linear equations, we construct an augmented matrix, and may modify it using the three elementary row operations:


*

*Interchange two rows.

*Multiply a row by a non-zero constant.

*Add a row to another row.


These correspond to operations on the system of equations.  It's fairly clear that the first two operations preserve the set of solutions.  It is also clear that any solution to the original system of equations will also satisfy to the modified system of equations.
Question:  How do we know that any solution to the modified system of equations also satisfies the original system of equations?

Every of these three operations has the inverse, which has the same form. So apply the inverses to the modified system and use your reasonings. E.g., you know that multipling a row by $a\ne 0$ preserves the set of solutions. So multiply the row of the modified system by $a^{-1}\ne 0$. Then you get the original system, and the set of solutions is preserved.
