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I have the following matrix: $$A= \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}$$

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

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.

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  • $\begingroup$ Sorry about the people who vote close without giving a reason. I guess their problem is that you did not specify the exact model. $A$ is an input-output matrix and you switched the first two vectors, right? $\endgroup$ – Giskard Oct 26 '15 at 7:27
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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&2\\2&3\end{bmatrix}\end{equation}

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

yields \begin{equation}B' = \begin{bmatrix}1&2&3\\0&1&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.

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    $\begingroup$ While this is perfectly true, the OP asked for economic interpretation, not an explanation of the mathematical operation. $\endgroup$ – Giskard Oct 26 '15 at 7:27
  • $\begingroup$ Hmm if my intuition did not badly mislead me, if we pivot the input output matrix we should obtain the quantities used in the production process to arrive at the given output vector. $\endgroup$ – HRSE Oct 26 '15 at 13:59
  • $\begingroup$ @HREcon Shouldn't your comment be under the question rather than the answer? And I think the problem with your comment is that it is not a 'full pivot' we are seeing so you probably have to alter the interpretation a little. $\endgroup$ – Giskard Oct 26 '15 at 15:56
  • $\begingroup$ @denesp I didn't give a mathematical explanation... I guess this is the more economic intuition that a pivot can have. More economic intuition could be derive from example of a maximization problem. The only way I can think about economic intuition would be that : X is the vector of quantities in good 1 and good 2, and when you solve a maximization problem (either a producer-type or consumer-type) you end up with first order conditions in the form a linear system, and then using pivot just the way I described it would yield optimal quantities. Should I edit my answer in that sense? $\endgroup$ – Louis. B Oct 26 '15 at 18:06
  • $\begingroup$ This is entirely up to you, but your last comment still did not contain a single economic reference. (Prices, production, inputs, etc.) $\endgroup$ – Giskard Oct 26 '15 at 18:12

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