Row Reduced Echelon Matrix

Last updated Nov 1, 2022

Let $F$ be a Field and let $m, n \in \mathbb{N}$. Then $R \in F^{m \times n}$ is a Row Reduced Echelon Matrix if

1. $R$ is a Row Reduced Matrix,
2. If $R_{i \cdot} = \mathbf{0}$ for $i \in [m]$, then $\not\exists k > i$ so that $R_{k \cdot} \neq \mathbf{0}$.
3. Let $r = \max\limits {i \in [m] : R_{i \cdot} \neq \mathbf{0}}$. Then for $k_{i} = \min\limits{j \in [n] : R_{ij} \neq 0}$, we have that $k_{1} < \cdots < k_{r}$.

1. The Identity Matrix.
2. $$\begin{pmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \end{pmatrix}$$
3. $$\begin{pmatrix}1 & 3 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

1. $$\begin{pmatrix}0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \\ 1 & 3 & 0 & 2 \end{pmatrix}$$ This is a Row Reduced Matrix, but it violates (2) at row 2 and (3) between row 1 and row 3

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