What is the interpretation of the output matrix of pivoting?
Pivoting helps solving linear system of equations. Imagine you want to solve for X that satisfies :

\begin{equation}A\cdot X = Y\end{equation}

where 
\begin{equation}X = \begin{bmatrix}x_1\\x_2\end{bmatrix}\end{equation} and \begin{equation}Y = \begin{bmatrix}y_1\\y_2\end{bmatrix}=\begin{bmatrix}3\\4\end{bmatrix}\end{equation} and \begin{equation}A = \begin{bmatrix}1&amp;2\\2&amp;3\end{bmatrix}\end{equation}

Then if applying Gauss' Pivot to A' (concatenation of A and Y) with 
\begin{equation}A' = \begin{bmatrix}1&amp;2&amp;3\\2&amp;3&amp;4\end{bmatrix}\end{equation}

yields 
\begin{equation}B' = \begin{bmatrix}1&amp;2&amp;3\\0&amp;1&amp;2\end{bmatrix}\end{equation}

you instantly have that $x_2=2$ and $x_1=3-2x_2=-1$. On large linear system it is very fast to solve them that way.