Can we specify all row equivalent matrices of a given matrix? Say we have a RREF matrix like $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}$$
From this matrix, is there some way of specifying ALL of the matrices which are row equivalent to this one?
 A: Doing an elementary row operation on a matrix is equivalent to multiply the matrix (on the left) by a particular invertible matrix (usually called an elementary matrix). So the elimination on a matrix $A$ to a RREF matrix $U$ is the same as writing $FA=U$, where $F$ is the product of the elementary matrices used in the process.
Any invertible matrix can be expressed as the product of elementary matrices: just reduce the matrix to its RREF, which is the identity and apply the same as before.
Thus the matrices that are row equivalent to $U$ are all matrices of the form $XU$, with $X$ invertible. If $X=[x_1\ x_2\ x_3\ x_4]$ ($x_i$ is the $i$-th column of $X$) and we write your matrix $U=[e_1\ 0\ e_2\ 0]$, then
$$
XU=[Xe_1\ X0\ Xe_2\ X0]=[x_1\ 0\ x_2\ 0]
$$
Since $X$ is an arbitrary invertible matrix, its columns $x_1$ and $x_2$ are arbitrary linearly independent vectors.
A: Hint:
$$\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&0&0&0\end{bmatrix}$$
Has vectors $(x_1,0,0,0)$ and $(0,0,x_3,0)$, which lets us have any linear combination of $(\alpha x_1,0,\beta x_3,0)$ with arbitrary $\alpha,\beta$.
