Gaussian Elimination and Row Echelon Form

The three row echelon forms that can be obtained directly are illustrated in the following examples.

$\blacktriangleright$ Matrices + Fraction-free Gaussian Elimination

$\left[\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right.$$\begin{array}{cc}
a & b \\
c & d
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right]$, fraction-free Gaussian elimination: $\left[\vphantom{
\begin{array}{cc}
a & b \\
0 & da-bc
\end{array}
}\right.$$\begin{array}{cc}
a & b \\
0 & da-bc
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a & b \\
0 & da-bc
\end{array}
}\right]$

$\left[\vphantom{
\begin{array}{rrr}
8 & 2 & 3 \\
2 & -5 & 8
\end{array}
}\right.$$\begin{array}{rrr}
8 & 2 & 3 \\
2 & -5 & 8
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
8 & 2 & 3 \\
2 & -5 & 8
\end{array}
}\right]$, fraction-free Gaussian elimination: $\left[\vphantom{
\begin{array}{rrr}
8 & 2 & 3 \\
0 & -44 & 58
\end{array}
}\right.$$\begin{array}{rrr}
8 & 2 & 3 \\
0 & -44 & 58
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
8 & 2 & 3 \\
0 & -44 & 58
\end{array}
}\right]$

$\blacktriangleright$ Matrices + Reduced Row Echelon Form

$\left[\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right.$$\begin{array}{cc}
a & b \\
c & d
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right]$, Reduced row echelon form: $\left[\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
}\right.$$\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
}\right]$

$\left[\vphantom{
\begin{array}{rrr}
8 & 2 & 3 \\
2 & -5 & 8
\end{array}
}\right.$$\begin{array}{rrr}
8 & 2 & 3 \\
2 & -5 & 8
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
8 & 2 & 3 \\
2 & -5 & 8
\end{array}
}\right]$, Reduced row echelon form: $\left[\vphantom{
\begin{array}{ccr}
1 & 0 &
\frac{31}{44}\vspace{8pt} \\
0 & 1 & -\frac{29}{22}
\end{array}
}\right.$$\begin{array}{ccr}
1 & 0 &
\frac{31}{44}\vspace{8pt} \\
0 & 1 & -\frac{29}{22}
\end{array}$$\left.\vphantom{
\begin{array}{ccr}
1 & 0 &
\frac{31}{44}\vspace{8pt} \\
0 & 1 & -\frac{29}{22}
\end{array}
}\right]$