Equivalent matrices

Two matrices $$\mathbf A$$ and $$\mathbf B$$ are called equivalent matrices (shown as $$\mathbf A\sim \mathbf B$$) if one can be derived by the other using only left or right multiplication by elementary matrices (p134 in ). In other words

$$\mathbf A \sim \mathbf B \iff \mathbf{B=PAQ}$$ for nonsingular $$\mathbf P$$ and $$\mathbf Q$$.

Row and column equivalence
Two matrices are called
 * row equivalent if $$\mathbf B = \mathbf{PA}$$ for some nonsingular $$\mathbf P$$, and
 * column equivalent if $$\mathbf B = \mathbf{AQ}$$ for nonsingular $$\mathbf Q$$.