# 1  Row Echelon Forms

Given a matrix $A$. In what follows a nonzero row (nonzero column) is a row (column, respectively) which contains at least one nonzero element. A leading entry of a nonzero row is the leftmost nonzero entry, i.e. the element ${a}_{i,j}\ne 0$ such that all ${a}_{i,k}=0$ for every $k.

Definition 1.1. Given a matrix $A$. We say that $A$ is in echelon form (or row echelon form) if it satisfies the following properties:

1. All nonzero rows are above any zero row.
2. Each leading entry is to the right of any leading entry of the rows above it. (I.e. if ${a}_{i,j}$ and ${a}_{r,s}\phantom{\rule{mediummathspace}{0.2em}}$are leading entries of their respective rows and if $r then $s.)
3. All entries in the column of a leading entry bellow it are zero. (I.e. if ${a}_{i,j}$ is the leading entry of the $i$-th row then ${a}_{k,j\phantom{\rule{mediummathspace}{0.2em}}}=0$ for every $k>i$.)

We say that the matrix $A$ is in reduced echelon form (or reduced row echelon form) if moreover

1. The leading entry in every nonzero row equals 1.
2. Each leading entry is the only nonzero entry in its column. (I.e. if${a}_{i,j}=1\phantom{\rule{mediummathspace}{0.2em}}$is the leading entry of $i$-th row then ${a}_{k,j}=0$ for every $k\ne i$.)

Leading entries are sometimes also called pivots.

Example The following matrices $A,\phantom{\rule{mediummathspace}{0.2em}}B\phantom{\rule{mediummathspace}{0.2em}},\phantom{\rule{mediummathspace}{0.2em}}D\phantom{\rule{mediummathspace}{0.2em}}$are in row echelon form, while the matrix $C$ is not; matrix $D$ is in reduced row echelon form.

$A\phantom{\rule{mediummathspace}{0.2em}}=$ $\left(\begin{array}{cccccc}2& 4& -1& 2& -4& 0\\ 0& 0& 3& 1& 2& -4\\ 0& 0& 0& 1& 0& 5\\ 0& 0& 0& 0& 0& 0\end{array}\right),$

 $B=\left(\begin{array}{cccccc}1& 0& 0& 0& 4& 6\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 7& 8\\ 0& 0& 0& 0& 0& 1\end{array}\right)\phantom{\rule{mediummathspace}{0.2em}},$

$C\phantom{\rule{mediummathspace}{0.2em}}=\phantom{\rule{mediummathspace}{0.2em}}\left(\begin{array}{cccccc}\phantom{\rule{mediummathspace}{0.2em}}-1& 0& 4& 0& 3& -1\\ 0& 1& 3& 0& 6& 0\\ 0& 0& 0& 0& 1& 4\\ 0& 0& 4& 6& 1& -1\end{array}\right),$

 $D=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 6\\ 0& 0& 1& 0& 0& -1\\ 0& 0& 0& 1& 0& 8\\ 0& 0& 0& 0& 1& -7\end{array}\right)$

Note that if we interchange the fourth and third row in the matrix $C$

we also get a matrix in row echelon form.

Remark We can say that any matrix of the form

$A\phantom{\rule{mediummathspace}{0.2em}}=\phantom{\rule{mediummathspace}{0.2em}}\left(\begin{array}{cccccc}\ast & \circ & \circ & \circ & \circ & \circ \\ 0& \ast & \circ & \circ & \circ & \circ \\ 0& 0& 0& \ast & \circ & \circ \\ 0& 0& 0& 0& \ast & \circ \end{array}\right)\phantom{\rule{mediummathspace}{0.2em}}\mathrm{or}$ $B\phantom{\rule{mediummathspace}{0.2em}}=\phantom{\rule{mediummathspace}{0.2em}}\left(\begin{array}{cccccc}0& \ast & \circ & \circ & \circ & \circ \\ 0& 0& \ast & \circ & \circ & \circ \\ 0& 0& 0& \ast & \circ & \circ \\ 0& 0& 0& 0& \ast & \circ \end{array}\right)$

where $\ast$ is any nonzero entry, $\circ$ is any entry (zero or nonzero)

is in row echelon form.

Any matrix

$C\phantom{\rule{mediummathspace}{0.2em}}=\phantom{\rule{mediummathspace}{0.2em}}\left(\begin{array}{ccccccc}1& 0& \circ & 0& 0& \circ & \circ \\ 0& 1& \circ & 0& 0& \circ & \circ \\ 0& 0& 0& 1& 0& \circ & \circ \\ 0& 0& 0& 0& 1& \circ & \circ \end{array}\right)$

where $\circ$ is any entry (zero or nonzero), is in the reduced row echelon form.