The procedure for GuassJordan Elimination is as follows:
 Find the leftmost column that is not all zeros and swap its row with the top row.
 Make the leading entry in the top row a "$1$". (If the top entry is $a$, then multiply the top row by $1/a$).
 Use the top row to make all the other entries in the column containing the leading one into zeros.
 Ignoring the top row, repeat steps 1 to 4 until there are no more leading ones.
Example:
Solve for $x$, $y$, and $z$ in:$ \begin{eqnarray} 2y  3z &= 2\cr 2x + z &= 3\cr x  y + 3z &= 1\cr \end{eqnarray} $Solution:Now we can read off the answer: $x = 5/4$, $y = 7/4$, and $z = 1/2$.
$$ \Matrix{ 0& 2& 3& & 2\cr 2& 0& 1& \& 3\cr 1& 1& 3& & 1\cr } $$ write the system as an augmented matrix $$ \Matrix{ 1& 1& 3& & 1\cr 2& 0& 1& \& 3\cr 0& 2& 3& & 2\cr } $$ interchange first and third rows
(to make top left entry nonzero)$$ \Matrix{ 1& 1& 3& & 1\cr 0& 2& 5& \& 1\cr 0& 2& 3& & 2\cr } $$ Add $2$ times first row to second row
(to get $0$ in first column of row 2)$$ \Matrix{ 1& 1& 3& & 1\cr 0& 1& 5/2& \& 1/2\cr 0& 2& 3& & 2\cr } $$ Divide second row by $2$
(to get a leading $1$ in row 2)$$ \Matrix{ 1& 0& 1/2& & 3/2\cr 0& 1& 5/2& \& 1/2\cr 0& 0& 2& & 1\cr } $$ Add second row to first row
Add $2$ times second row to third
(to get $0$'s in the second column)$$ \Matrix{ 1& 0& 1/2& & 3/2\cr 0& 1& 5/2& \& 1/2\cr 0& 0& 1& & 1/2\cr } $$ Divide third row by 2
(to get a leading $1$)$$ \Matrix{ 1& 0& 0& & 5/4\cr 0& 1& 0& \& 7/4\cr 0& 0& 1& & 1/2\cr } $$ Add $1/2$ times third row to first row
Add $5/2$ times third row to second row
(to get $0$'s in third column)Check that the answer satisfies the initial equations (in case we made arithmatic errors):
$ \begin{array}{l} 2y  3z = 2(7/4)  3(1/2) = 7/2  3/2 = 4/2 = 2\cr 2x + z = 2(5/4) + 1/2 = 5/2 + 1/2 = 6/2 = 3\cr x  y + 3z = (5/4)  (7/4) + 3(1/2) = (2/4) + 3/2 = (1/2) + 3/2 = 2/2 = 1\cr \end{array} $All of these check out, so our solution is correct.

