TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The determinant differential is $D \det (A) [H] = tr (adj (A) H)$ at every matrix, and Jacobi's formula holds on the invertible locus

Statement

Let $A$ be an $n \times n$ matrix and let $H$ be a direction matrix. Then

$D \det (A) [H] = tr (adj (A) H) .$

If $A$ is invertible, then

$D \det (A) [H] = \det (A) tr (A^{- 1} H) .$

Facts & Assumptions

Given: An $n \times n$ matrix $A$ and a perturbation direction $H$ .

[L1]

On the invertible locus, the derivative of inversion is $D inv (A) [H] = - A^{- 1} H A^{- 1}$ (On the invertible locus, $D inv (A) [H] = - A^{- 1} H A^{- 1}$ ).

Proof

technique · direct

1.1

The determinant is multilinear in the columns. In $\det (A + t H)$ , the coefficient of $t$ comes from replacing exactly one column of $A$ by the corresponding column of $H$ . Those cofactors are precisely the entries of $adj (A)$ , so $D \det (A) [H] = tr (adj (A) H)$ .

givenalgebra

2.1

If $A$ is invertible, then $adj (A) = \det (A) A^{- 1}$ . Substituting this identity into step 1.1 gives $D \det (A) [H] = \det (A) tr (A^{- 1} H)$ , which is Jacobi's formula on the invertible locus. The domain restriction matters because $A^{- 1}$ appears explicitly there, unlike the adjugate formula of step 1.1.

L1step 1.1algebra∎

Depends on

Used by

The adjugate formula still differentiates the determinant at a singular matrix, while Jacobi's inverse form does not Example

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alan Edelman and Steven G. Johnson, Matrix Calculus for Machine Learning and Beyond (standard reference, not scraped)

The determinant differential is Ddet⁡(A)[H]=tr⁡(adj⁡(A)H)Ddet(A)[H]=tr(adj(A)H) at every matrix, and Jacobi's formula holds on the invertible locus