Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13
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Positive determinant does not imply positive definiteness

Statement refuted

Refuted claim: Every real symmetric matrix with positive determinant is positive definite.

Facts & Assumptions

Given: The real diagonal matrix A=diag⁡(−1,−1,1).

[L1]

Positive definiteness requires vTAv>0 for every nonzero v, and the signs of a diagonal form give its inertia (Positive and negative definiteness, the inertia (p,q,r), rank p+q, and signature p−q of a real symmetric bilinear or quadratic form).

[L2]

The determinant is the signed Leibniz sum, which for a diagonal matrix reduces to the product of its diagonal entries (For n≥1, the determinant over a commutative ring by the Leibniz formula, and ∣det⁡A∣ for a real matrix).

Counterexample

technique · direct computation
1.1

By [L2], det⁡A=(−1)(−1)(1)=1>0.

L2algebra
1.2

But e1TAe1=−1<0, so [L1] shows that A is not positive definite. Its inertia is (1,2,0).

L1algebra
2.1

Thus a positive determinant alone does not imply positive definiteness.

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 57 results over 10 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