Row Operation has Inverse

Theorem
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$.

Let $\mathbf A \in \map \MM {m, n}$ be a matrix.

Let $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.

Then there exists another elementary row operation $\Gamma'$ which transforms $\mathbf B$ back to $\mathbf A$.

Proof
Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the finite sequence of elementary row operations that compose $\Gamma$.

Let $\mathbf A_1$ denote the matrix which results from applying $e_1$ to $\mathbf A$.

For each $i \in \set {2, 3, \ldots, k}$, let $\mathbf A_i$ denote the matrix which results from applying $e_i$ to $\mathbf A_{i - 1}$.

Thus we have that $\mathbf A_k = \mathbf B$.

From Elementary Row Operation has Inverse, each of $e_i$ has a corresponding elementary row operation $e'_i$ which transforms $\mathbf A_i$ back to $\mathbf A_{i - 1}$.

Thus let $\sequence {e'_j}_{1 \mathop \le j \mathop \le k}$ be the finite sequence of elementary row operations defined as:


 * $e'_j = e_{k - i + 1}$

Let $\Gamma'$ be the row operation composed of the finite sequence of elementary row operations $\sequence {e'_j}_{1 \mathop \le j \mathop \le k}$.

Thus $\Gamma'$ is a row operation which transforms $\mathbf B$ into $\mathbf A$.

Hence the result.