Differentiate the Gaussian elimination and LU- Factorization in solving system of linear equations.

Differentiate the Gaussian elimination and LU- Factorization in solving system of linear equations.
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nitruraviX

Let the system of linear equation be
$AX=b$
Gauss elimination for system of linear equation by back substitution for the system give the upper triangular system of linear equation which is solvable easily.
If we apply Gauss elimination for the matrix of the the system then it gives the equivalent matrix which is the product of two matrices, upper triangular matrix U and Lower triangular matrix L.
i.e.,
$A=LU$.
then the system is
$AX=b$
$LUX=b$
$UX={L}^{-1}b$
$x={U}^{-1}\left({L}^{-1}b\right)$