The reduced row-echelon form

We can say the matrix as reduced row-echelon form if following four conditions are satisfied :

1. If a row does not have entirely zero components, then the first nonzero element in the row must be 1. (This is called "leading 1")

2. If a row does have entirely zero components, the row(s) is arranged to the bottom of the matrix.

3. the row consisted of leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

4. Each column that contains a leading 1 has zeros everywhere else in that column.

for example following matrices are in reduced row-echelon form :

Gamma function

$\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\]$

or

[\Gamma(z) = \int_0^1 \left( \ln(\frac{1}{t}) \right)^{z-1}dt]