What is the interpretation of the output matrix of pivoting? I have the following matrix:
$$A=  \begin{bmatrix}
    1 &amp; 2 &amp; 3 \\
    2 &amp; 3 &amp; 4 
  \end{bmatrix}$$

After pivoting, I got this matrix:
$$B=  \begin{bmatrix}
    1 &amp; 2 &amp; 3 \\
    0 &amp; -1 &amp; -2 
  \end{bmatrix}$$

It's very embarrassing, but I know how to pivot, but I don't understand the interpretation of this matrix operation. What is pivoting and why is it important for linear optimization in economics? I only found resources on the mathematical intuition of pivot operation.

In the meantime, thank you so much for your attention and participation.
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.