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Every real $m\times n$ matrix $A$ can be factored in three ways that arise from the steps of elimination: a lower triangular/upper triangular factorization $PA=LU$, a column-row factorization $A=CR$, and a triple factorization $A=C{W}^{-1}B$. The column-row factorization provides both a constructive proof that the row rank $r$ of $A$ equals the column rank, and a formula for the pseudoinverse $A^{+}$ not based on the singular value decomposition. In the triple factorization, $C$ and $B$ contain the first $r$ independent columns of $A$ and the first $r$ independent rows of $A$; $W$ is the invertible submatrix of $A$ where $B$ meets $C$. An alternative to the traditional elimination method, using slide steps in place of the usual swap steps, identifies the first $r$ independent rows of $A$.