Row equivalence

Definition

Let $F$ be a field and let $A$ and $B$ be $m \times n$ matrices over $F$. The matrix $A$ is row equivalent to $B$ when there is a finite sequence of elementary row operations that changes $A$ into $B$.

Remarks

Row equivalence is an equivalence relation, because each elementary row operation has an inverse operation. The matrices $A$ and $B$ are row equivalent if and only if $B = PA$ for an invertible $m \times m$ matrix $P$. Each matrix is row equivalent to exactly one matrix in reduced row echelon form. Row equivalent matrices have the same row space and the same null space.

Explanations

L1 human

Two matrices are row equivalent when row operations can turn one into the other. If they are augmented matrices of two systems, then the two systems have the same solutions. Row reduction replaces a matrix by a simpler row equivalent matrix.