False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.
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FALSE: $\det(A+B)=\det(A)+\det(B)$ for all same-sized square matrices

Statement

The following claim is false: for all same-sized square real matrices, $\det(A+B)=\det(A)+\det(B).$

Facts & Assumptions

Given: Over $\mathbb R$ , take $A=B=I_2$ .

[L1]

Matrix addition is entrywise and $I_2$ has diagonal entries $1$ (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).

[L2]

Determinant is multilinear in one selected row or column while all other rows or columns are fixed (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring).

[L3]

For a $2\times2$ matrix, determinant is $ad-bc$ (The Leibniz formula gives $\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$ ).

[L4]

The real numbers form a field (The reals form a field).

Refutation

technique · direct

1.1

Here $A+B=2I_2$ , so [L3] gives $\det(A+B)=4$ , whereas $\det(A)+\det(B)=1+1=2$ . Since $4\ne2$ in $\mathbb R$ , the claim fails.

L1L3L4algebra

2.1

The property in [L2] does not imply whole-matrix additivity: changing both rows at once produces cross terms. It permits addition only in one selected row or column while all the others remain fixed.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 13 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

Purdue University, 3.1 + 3.2 Determinants, Properties of Determinants (standard reference, not scraped)
D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)

FALSE: det⁡(A+B)=det⁡(A)+det⁡(B)\det(A+B)=\det(A)+\det(B)det(A+B)=det(A)+det(B) for all same-sized square matrices