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For completeness, we have added the following content to the Exercises section of the Matrix Inverter tool available at

http://www.minerazzi.com/tools/matrix-inverter/gauss-jordan.php

and mentioned in the post

https://irthoughts.wordpress.com/2016/10/05/matrix-inverter-a-matrix-inversion-tool/

The following information was found online (Quora, 2013, StackExchange, 2013a; 2013b).

Let Ʃ be a covariance matrix and Ʃ-1 an inverse covariance matrix, commonly referred to as the precision matrix.

With Ʃ, one observes the unconditional correlation between a variable i, to a variable j by reading off the (i,j)-th index.

It may be the case that the two variables are correlated, but do not directly depend on each other, and another variable k explains their correlation. By computing Ʃ-1 we can examine if the variables are partially correlated and conditionally independent.

Ʃ-1 displays information about the partial correlations of variables. A partial correlation describes the correlation between variable i and j, once you condition on all other variables. If i and j are conditionally independent then the (i,j)-th element of Ʃ-1 will equal zero. If the data follows a multivariate normal then the converse is true, a zero element implies conditional independence.

In general, Ʃ-1 is a measure of how tightly clustered the variables are around the mean (diagonal elements) and the extend to which they do not co-vary with the other variables (non-diagonal elements). The higher the diagonal elements, the tighter the variables are clustered around the mean.

So far I found that to be, in my opinion, the simplest explanation on the subject. So there you have a good application for our Matrix Inverter tool.

References