Use of elementary matrices

Question

What are the advantages of using elementary matrices to solve a system of linear equations? Why would anyone even try to find an elementary matrix when you can just multiply and add the columns of a given system and then use back-substitution?

Answer 1

Using elementary matrices is the same as just multiplying an adding the columns (or rows). The point is that the former is the formal way of defining these operations, while still staying inside the paradigm of multiplying matrices.

Quite possibly it would be much less of an effort to describe certain algorithms using elementary matrices as well.

Answer 2

Elimination, which is usually called Gaussian elimination, can be viewed in many different ways. The simple operations with rows is one way. Multiplication by elimination matrices (more typically called Gauss transforms) is another. Yet another one is to view it as LU factorization (given $ A $, the computation of unit lower triangular matrix $ L $ and upper triangular matrix $ U $ so that $ A = L U $). In different situations viewing the algorithms in different ways is beneficial. For example, if you view it as the application of a sequence of elimination matrices, it is easy to show how (Gaussian) elimination and LU factorization are one and the same.

In practice, elimination is computed as an LU factorization. The reason is that if you have $ A = LU $ then given a right-hand side, you can solve $ A x = b $ by solving $ L z = b $ followed by solving $ U x = z $. Of course, not all matrices have an LU factorization, which then brings in (partial) pivoting - row swapping - which is also more easily described as the application of permutation matrices intertwined with Gauss transforms.

Anyway, I can go on and on... If you want a reference, let me know.