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I am studying the LU factorisation. What I have learned is that with this technique we start with a matrix $A$ and result into two matrices $L$ and $U$ where $L$ is a Lower Triangular matrix and $U$ is an Upper Triangular matrix.

One property of the $L$ and $U$, is that when you multiple $L$ and $U$ you should result to the original matrix $A$.

Also the $L$ matrix should have $1$s to its leading diagonal.

All that listens goods, but I can't understand where this is useful? What is the purpose of doing that? Is that Alternative of the Gaussian Elimination ?

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The $LU$ factorization is in fact Gaussian elimination where you store intermediate coefficients where you made zeroes appear. Then benefit is that you can solve the same system with different RHS or compute the matrix inverse without repeating some calculations and without extra storage.

To solve a single system with a single RHS, there is no particular benefit.

(You will also learn later that linear algebraists love to express matrix operations in factorized forms.)

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