### Row
Operation

Exercise Prove that if a matrix **A** can be changed to a matrix
**B** by row operations, then the matrix **B** can also be changed to a matrix
**A** by row operations.

Answer
If applying `r`[row `i`] + [row `j`] to **A** gives us **B**,
then applying (-`r`)[row `i`] + [row `j`] to **B** gives us **A** back.

If applying [row `i`] ↔ [row `j`] to **A** gives us **B**,
then applying the same operations again to **B** gives us **A** back.

If applying `d`[row `i`] to **A** gives us **B**,
then applying `d`^{-1}[row `i`] to **B** gives us **A** back.