Row Reduced Matrix

Last updated Nov 1, 2022

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

1. $\forall i \in [m]$, the leading nonzero entry of $R_{i \cdot}$ is $1$. That is, for $j = \min\limits{j’ \in [n] : R_{ij’} \neq 0}$, $R_{ij} = 1$. This is vacuously true if ${j’ \in [n] : R_{ij’} \neq 0} = \emptyset$.
2. $\forall j \in [n]$, if $R_{\cdot j}$ contains the leading nonzero entry of some row, then all other entries of $R_{\cdot j} = 0$. This is, if $\exists i \in [m]$ so that $j = \min\limits{j’ \in [n] : R_{ij’} \neq 0}$, then $R_{kj} = 0$ $\forall k \neq i$.

1. The Identity Matrix $$I_{ij} = \delta_{ij} = \begin{cases} 1 & \text{if } i = 1 \\ 0 & \text{otherwise}\end{cases}$$
2. $$\begin{pmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \end{pmatrix}$$
3. $$\begin{pmatrix}0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \\ 1 & 3 & 0 & 2 \end{pmatrix}$$

1. $$\begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 0\end{pmatrix}$$ This matrix violates the 2nd defining property on column 2.
2. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2\end{pmatrix}$$ This matrix violates the 1st defining property on row 3.