Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4

Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
Let $S$ be a system of homogeneous simultaneous linear equations:


 * $\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$

Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.

Let the number of non-zero rows of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be $l$.

If $l = n$, then the only solution to $S$ is the trivial solution.

Proof
Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.

Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_n$.

As there are $n$ columns as well as $n$ non-zero rows:
 * each row has exactly one $1$ in it
 * each column has exactly one $1$ in it.

Thus $S'$ can be expressed as:

and the result follows.