# how to find inverse of a nxn matrix when n is large ie n=8,10...

how to find inverse of a nxn matrix when n is large ie n=8,10...
please provide proper method and example

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Commented:
You typically use Gaussian Elimination to invert matrices:

http://mathworld.wolfram.com/GaussianElimination.html

http://en.wikipedia.org/wiki/Gaussian_elimination

It works, you can do it by hand, but it is not likely to be much fun.
Even in Linear Algebra courses, they don't make you work with anything over 4x4.

There are solvers on line that will do up to 30x30.

http://www.bluebit.gr/matrix-calculator/

Matlab is probably the software of choice for this sort of thing.
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Commented:
The Numerical Recipes series of books are good for learning about this sort of thing.  There used to be free PDF's floating around the internet, but now they're all hosted on virus-laden .ru sites and the current publisher's horribly encumbered site.

There's a lot of drama about the unique license for the code in the book, see http://www.astro.umd.edu/~bjw/software/boycottnr.html for details.  So don't include any Numerical Recipes code in a project that is potentially vulnerable to a lawsuit.

But if you learn the right chapter in the book, you will understand what is required to write your own good inversion routine, and how to check that your routine works.

Just make sure whatever you use has pivoting.
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Commented:
You might also look at the Gauss-Seidel method (http://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method) and the "over relaxation" techniques for improving convergance (http://en.wikipedia.org/wiki/Successive_over-relaxation). Note also that practically, very large matrixes  are often full of zeros, so there are various "banding" techniques which can improve performace.

Gaussian elimination will always, when such exists, produce a result, although there are two problems associated with it. 1) the number of operatuions is n³, so when n is large this is a very lage number of operations. The Seidel method tends to n² so is somewhat faster. 2) One can get very large or very small numbers as intermediate results, which propates inaccuracies into the result. The other methods tend to avoid this.
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Commented:
Enough information has been provided.
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Commented:
Split point between all three responders.
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Commented:
Split point between all three responders.

d-glitch   http:#a24259952  167 points
Cited the relevant Wikipedia article and an on-line calculator that is still working seven years later.

Cited Numerical Recipes, probably the best text practical text on the subject.  His link has gone stale.

BigRat   http:#a24268295   167 points
Cited Wikipedia articles that provided more depth on the subject and valuable information on faster algorithms.
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