Row Reduction and Echelon Forms

Definitions §

leading entry: leftmost non-zero entry of row

echelon form:

1. all non-zero rows are above all-zero rows
2. leading entry of row is to the right of leading entry of column above
3. all entries in column below leading entry are 0

reduced echelon matrix:

1. echelon form
2. leading entry in each nonzero row is 1

Example Problem §

Express in row-echelon form

$$\begin{array}{r}\left(\begin{array}{ccccc}0& -3& -6& 4& 9\\ -1& -2& -1& 3& 1\\ -2& -3& 0& 3& -1\\ 1& 4& 5& -9& -7\end{array}\right)\end{array}$$ $$\begin{array}{rl}{R}_1\leftrightarrow R_3,R_2\to \frac{1}{2}(R_2+R_1),R_3\to \frac{1}{5}(R_3+2R_1)& ⟹\left(\begin{array}{ccccc}1& 4& 5& -9& -7\\ 0& 1& 2& -3& -3\\ 0& 1& 2& -3& -3\\ 0& -3& -6& 4& 9\end{array}\right)\\ R_3\to \frac{1}{5}(R_3+3R_2)& ⟹\left(\begin{array}{ccccc}1& 4& 5& -9& -7\\ 0& 1& 2& -3& -3\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right)\\ R_2\to R_2+3R_3,R_1\to R_1-4R_4,R_1\to R_1-3R_3& ⟹\left(\begin{array}{ccccc}1& 0& -3& 0& 5\\ 0& 1& 2& 0& -3\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right)\end{array}$$