ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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A full $3\times3$ Leibniz expansion lists all six permutations and their signs

Example

For $A=(a_{ij})\in M_3(R)$ , $\det A=a_{00}a_{11}a_{22}+a_{10}a_{21}a_{02}+a_{20}a_{01}a_{12}-a_{10}a_{01}a_{22}-a_{20}a_{11}a_{02}-a_{00}a_{21}a_{12}.$

Facts & Assumptions

Given: A $3\times3$ matrix over a commutative ring.

[L1]

The determinant is the signed Leibniz sum (For $n\ge1$ , the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix).

[L2]

A permutation's sign is determined by the parity of its inversions (Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$ , and even and odd permutations).

[L3]

A set of size $3$ has $3!=6$ bijections to itself (A finite set $A$ with $\lvert A\rvert = n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality).

Verification

technique · direct

1.1

The even permutations in one-line notation are $[0,1,2]$ , $[1,2,0]$ and $[2,0,1]$ ; the odd ones are $[1,0,2]$ , $[2,1,0]$ and $[0,2,1]$ . These are all six permutations.

L1L2L3

2.1

Substitution yields the displayed six terms. For $A=\begin{pmatrix}1&2&3\\0&1&4\\5&6&0\end{pmatrix}$ they give $0+0+40-0-15-24=1$ , a concrete check of the signs.

step 1.1algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 18 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

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Example 4.27 (standard reference, not scraped)
P. Massot, Structures algébriques fondamentales, §6.4 (standard reference, not scraped)

A full 3×33\times33×3 Leibniz expansion lists all six permutations and their signs