ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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An invertible $3\times3$ real matrix factored explicitly into elementary matrices

Example

The matrix $A=\begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}$ has the elementary factorisation $A=\begin{pmatrix}1&0&0\\0&1&1\\0&0&1\end{pmatrix}\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}.$

Facts & Assumptions

Given: The displayed real matrix $A$ .

[L1]

Every invertible real square matrix is a finite product of elementary matrices (Every invertible finite square real matrix is a finite product of elementary matrices).

[L2]

A reduction $E_r\cdots E_1A=I$ reverses to an elementary factorisation (A finite row reduction from $A$ to $B$ is encoded by $B=E_r\cdots E_1A$ ).

Verification

technique · direct

1.1

Subtract row $2$ from row $1$ , then subtract the new row $1$ from row $0$ ; these elementary operations reduce $A$ to $I_3$ , so their matrices satisfy $E_2E_1A=I_3$ .

L1L2algebra

2.1

Reversing the operations gives the two displayed row-addition matrices. Their direct product is $A$ , verifying the factorisation entry by entry.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

An invertible 3×33\times33×3 real matrix factored explicitly into elementary matrices