Row Reduction as Multiplication by Elementary Matrices

A concept note on interpreting row reduction through elementary matrices, column motion and row geometry

Row reduction is often introduced as a sequence of row operations, but each row operation is also multiplication on the left by an elementary matrix. This turns Gaussian elimination into a product of simple invertible transformations.

The same algebraic sequence can be visualized in two complementary ways: by tracking how all columns move under left multiplication, or by tracking how row normal planes and row vectors change.

Explore more on GraphMath Linear Algebra

Column view

Left multiplication by an elementary matrix acts on every column of the matrix. In the column view, all columns move together as the row operations are applied.

Row normal planes and row vectors

Row operations directly change the equations represented by the rows. In the row view, row normal planes and row vectors show how each operation updates the row geometry.

Concept

If the elementary matrices are E₁, E₂, …, Eₖ, then row reduction can be written as R = E_k ··· E₂E₁A. The reduced matrix R is produced by multiplying the original matrix A on the left.

Since elementary matrices are invertible, the original matrix can also be rebuilt by reversing the row operations: A = E₁⁻¹E₂⁻¹ ··· E_k⁻¹R.

Key equations

R = Eₖ ··· E₂ E₁ A

A = E₁⁻¹ E₂⁻¹ ··· Eₖ⁻¹ R

For full-rank square A reduced to I: A = E₁⁻¹ E₂⁻¹ ··· Eₖ⁻¹ I

Elementary row operations = left multiplication by elementary matrices

These equations explain why a row-reduction algorithm can be interpreted as a sequence of geometric transformations.

Query phrases

row reduction as multiplication by elementary matrices
Gaussian elimination elementary matrices visualization
row operations as left multiplication
column view of row reduction
row normal planes row reduction visualization

References

Related chapter: GraphMath — Row reduction