How do I decompose a non-singular matrix [A], that is, how do I find ?

If forward elimination steps of Naïve Gauss elimination methods can be applied on a nonsingular matrix, then can be decomposed into LU as

1. The elements of the matrix are exactly the same as the coefficient matrix one obtains at the end of the forward elimination steps in Naïve Gauss Elimination.

2. The lower triangular matrix has 1 in its diagonal entries. The non zero elements on the non-diagonal elements in are multipliers that made the corresponding entries zero in the upper triangular matrix during forward elimination.

Let us look at this using the same example as used in Naïve Gaussian elimination.