Show that two matrices A and B are row equivalent iff there exists an invertible matrix C so that A=CB

Question

Show that two matrices $A,B\in \text{Mat}_{m,n}(\mathbb{F}$) are row equivalent iff there exists an invertible matrix $C\in \text{Mat}_{n}(\mathbb{F}$), so that $A=C\cdot B$.

Is the solution that $C$ is row equivalent with the identity matrix since $C$ is invertible? I'm still trying to grasp the basics 🙂

Answer 1

Any invertible matrix can be written as a composition of elementary matrices (each of which represents an elementary row operation). Now, apply the definition of row equivalence. This proves the statement.