Determinant Expansion Visualized by Permutations and Cofactors

A concept note on determinant calculation as signed products organized by permutations and cofactor expansion

The determinant can be expanded as a signed sum over permutations, and the same terms can also be organized by cofactor expansion. Each permutation contributes one signed product, while cofactor expansion groups those products by a chosen pivot and its remaining submatrix.

This visualization makes the selected entries, submatrices, permutation signs and cofactor grouping visible in the 3×3 and 4×4 cases.

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3×3 expansion by permutations and cofactors

In the 3×3 case, there are six permutation terms. The diagram shows the selected entries, their signs and the submatrices that appear when the same calculation is read as cofactor expansion.

Determinant expansion visualization for a 3 by 3 matrix, showing permutation terms, pivots, submatrices, cofactor signs and product terms.
Cropped preview of the 3×3 determinant expansion by permutations and cofactors. Click to open the full image.

4×4 expansion by permutations and cofactors

In the 4×4 case, the permutation formula produces twenty-four terms. The larger diagram shows how those signed products can be grouped through cofactor expansion into smaller determinant calculations.

Determinant expansion visualization for a 4 by 4 matrix, showing permutation terms, signs, pivots, submatrices and cofactor grouping.
Cropped preview of the 4×4 determinant expansion by permutations and cofactors. Click to open the full image.

Concept

For an n × n matrix, a permutation chooses one column index for each row. The condition that the chosen column indices form a permutation guarantees that no column is reused.

Cofactor expansion reorganizes the same determinant by fixing a pivot entry, deleting its row and column, and taking the determinant of the remaining submatrix. The cofactor sign supplies the alternating sign attached to that grouped set of permutation terms.

Key equations

det(A) = Σ sign(σ) a₁σ(1) a₂σ(2) ··· aₙσ(n)
σ ranges over all permutations of {1, 2, ..., n}
sign(σ) = +1 for even permutations
sign(σ) = −1 for odd permutations
Cᵢⱼ = (−1)ⁱ⁺ʲ det(Aᵢⱼ)
det(A) = aᵢ₁Cᵢ₁ + aᵢ₂Cᵢ₂ + ··· + aᵢₙCᵢₙ

The permutation formula lists all signed products directly. Cofactor expansion groups the same determinant terms through pivots and submatrices.

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References

Related chapter: GraphMath — Determinant