Row Equivalent Matrix Systems

Last updated Nov 1, 2022

Let $F$ be a Field. Let $m, n \in \mathbb{N}$, let $A, B \in F^{m \times n}$, and let $\mathbf{b}, \mathbf{c} \in F^{m \times 1}$. Then we say the Matrix Equation Systems $$\begin{align*} A \mathbf{x} &= \mathbf{b} \tag{1} \\ B \mathbf{x} &= \mathbf{c} \tag{2} \end{align*}$$are Row Equivalent Matrix Systems if $$\begin{pmatrix}A &\Big| &\mathbf{b}\end{pmatrix} \sim_{R} \begin{pmatrix}B &\Big| &\mathbf{c}\end{pmatrix}$$ where $(A| \mathbf{b}) \in F^{m \times n+1}$ represents the columnwise concatenation of $A$ and $\mathbf{b}$.

• Row Equivalent Matrices