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

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Differentiating the inverse of a $2 \times 2$ matrix reproduces the closed formula entrywise

Example

Take

$A = (\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array}), H = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$

Then

$A^{- 1} = (\begin{array}{cc} \frac{1}{2} & 0 \\ 0 & 1 \end{array}), - A^{- 1} H A^{- 1} = (\begin{array}{cc} - \frac{1}{4} & 0 \\ 0 & 1 \end{array}) .$

The direct first-order expansion of $(A + t H)^{- 1}$ has the same linear term.

Facts & Assumptions

Given: The displayed matrices $A$ and $H$ .

[L1]

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

Verification

technique · direct

1.1algebra

Since $A + t H = diag (2 + t, 1 - t)$ , one has $(A + t H)^{- 1} = diag ⁣ (\frac{1}{2 + t}, \frac{1}{1 - t}) = (\begin{array}{cc} \frac{1}{2} - \frac{1}{4} t & 0 \\ 0 & 1 + t \end{array}) + O (t^{2}) .$ Also direct multiplication gives $- A^{- 1} H A^{- 1} = (\begin{array}{cc} - \frac{1}{4} & 0 \\ 0 & 1 \end{array})$ .

2.1L1step 1.1algebra∎

Step 1.1 shows that the coefficient of $t$ in the exact inverse expansion is $(\begin{array}{cc} - \frac{1}{4} & 0 \\ 0 & 1 \end{array})$ , which is exactly $- A^{- 1} H A^{- 1}$ . This matches [L1] entry by entry.

Depends on

On the invertible locus, $D\operatorname{inv}(A)[H]=-A^{-1}HA^{-1}$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)

Differentiating the inverse of a 2×22×2 matrix reproduces the closed formula entrywise