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A finite row reduction from $A$ to $B$ is encoded by $B = E_{r} \dots E_{1} A$

Statement

Suppose a sequence of $r$ elementary row operations transforms $A$ into $B$ , and let $E_{1}, \dots, E_{r}$ be their elementary matrices in execution order. Then $B = E_{r} \dots E_{1} A .$ For $r = 0$ the empty product is the identity and $B = A$ .

Facts & Assumptions

Given: A finite sequence of elementary row operations taking $A$ to $B$ .

[L1]

One elementary row operation acts by left multiplication by its elementary matrix (Applying an elementary row operation is left multiplication by its elementary matrix).

[L2]

Compatible matrix multiplication is associative and has identity matrices (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

[L3]

The induction principle proves a property from its zero case and successor step (The principle of mathematical induction).

Proof

technique · induction

1.1

After no operations the intermediate matrix is $A = I_{n} A$ , the empty-product formula. If after $k$ operations it is $E_{k} \dots E_{1} A$ , then [L1] says the next operation gives $E_{k + 1} (E_{k} \dots E_{1} A)$ .

L1L3baseih

2.1

Associativity rewrites the successor expression as $E_{k + 1} E_{k} \dots E_{1} A$ ; induction therefore gives the formula after all $r$ operations, including $r = 0$ .

step 1.1L2discharge-induction∎

Depends on

Used by

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Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.3 (standard reference, not scraped)

A finite row reduction from AA to BB is encoded by B=Er⋯E1AB=Er​⋯E1​A