Elimination Matrices

See LinearAlgebra/Elimination for a walkthrough of elimination. This regards the computation of elimination matrices, which are a method of computing elimination.

Introduction

Consider the below system of equations:

x + 2y + z = 2
3x + 8y + z = 12
4y + z = 2

Formulation

The first step of elimination involves the elimination of the cell at row 2 column 1 (henceforward cell (2,1)).

[1] 2 1    [1] 2  1
 3  8 1 ->  0  2 -2
 0  4 1     0  4  1

This can instead be formulated in matrices:

┌       ┐┌      ┐   ┌       ┐
│  1 0 0││ 1 2 1│   │ 1 2  1│
│ -3 1 0││ 3 8 1│ = │ 0 2 -2│
│  0 0 1││ 0 4 1│   │ 0 4  1│
└       ┘└      ┘   └       ┘

This elimination matrix is called E_{2 1} because is eliminated cell (2,1). An elimination matrix is always the identity matric with the negative of the multiplier in the elimination position.

The full elimination process can be formulated as E_{3 2} (E_{2 1} A) = U. This is equivalent to (E_{3 2} E_{2 1}) A = U.

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