Consider the following Gauss-Jordan reduction:
$$\underbrace{\left[\begin{array}{ccc} 1 &0 &1\cr -6 &0 &-5\cr 0 &-6 &0 \end{array}\right]}_A \to \underbrace{\left[\begin{array}{ccc} 1 &0 &1\cr 0 &0 &1\cr 0 &-6 &0 \end{array}\right]}_{E_1 A} \to \underbrace{\left[\begin{array}{ccc} 1 &0 &1\cr 0 &-6 &0\cr 0 &0 &1 \end{array}\right]}_{E_2E_1 A} \to \underbrace{\left[\begin{array}{ccc} 1 &0 &1\cr 0 &1 &0\cr 0 &0 &1 \end{array}\right]}_{E_3E_2E_1 A} \to \underbrace{\left[\begin{array}{ccc} 1 &0 &0\cr 0 &1 &0\cr 0 &0 &1 \end{array}\right]}_{E_4E_3E_2E_1 A}=I$$
Find
$E_1 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $E_2 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $E_3 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$. $E_4 =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$.

Write $A$ as a product $A=E_1^{-1}E_2^{-1}E_3^{-1}E_4^{-1}$ of elementary matrices:

$\left[\begin{array}{ccc} 1 &0 &1\cr -6 &0 &-5\cr 0 &-6 &0 \end{array}\right] =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$