ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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

Example

Let

$H = (\begin{array}{cc} h_{11} & h_{12} \\ h_{21} & h_{22} \end{array}) .$

At the invertible matrix $I_{2}$ , one has $D \det (I_{2}) [H] = h_{11} + h_{22} = tr (H)$ . At the singular matrix $S = diag (1, 0)$ , one has

$\det (S + t H) = t h_{22} + O (t^{2}),$

so $D \det (S) [H] = h_{22} = tr (adj (S) H)$ , but Jacobi's formula cannot even be written because $S^{- 1}$ does not exist.

Facts & Assumptions

Given: The direction matrix $H$ , the identity $I_{2}$ , and the singular matrix $S = diag (1, 0)$ .

[L1]

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

Verification

technique · direct

1.1

At $A = I_{2}$ , one has $adj (I_{2}) = I_{2}$ , so [L1] gives $D \det (I_{2}) [H] = tr (H) = h_{11} + h_{22}$ .

L1algebra

2.1

At $S = diag (1, 0)$ , $\det (S + t H) = \det (\begin{array}{cc} 1 + t h_{11} & t h_{12} \\ t h_{21} & t h_{22} \end{array}) = t h_{22} + t^{2} (h_{11} h_{22} - h_{12} h_{21}),$ so $D \det (S) [H] = h_{22}$ . Since $adj (S) = diag (0, 1)$ , this is $tr (adj (S) H)$ , exactly as [L1] predicts. But $S^{- 1}$ does not exist, so Jacobi's inverse-locus formula is unavailable.

L1algebra∎

Depends on

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)