False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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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 $R$ , 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 $2 \times 2$ matrix, determinant is $a d - b c$ (The Leibniz formula gives $\det (\begin{array}{cc} a & b \\ c & d \end{array}) = a d - b c$ ).

[L4]

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

Refutation

technique · direct

1.1L1L3L4algebra

Here $A + B = 2 I_{2}$ , so [L3] gives $\det (A + B) = 4$ , whereas $\det (A) + \det (B) = 1 + 1 = 2$ . Since $4 \neq 2$ in $R$ , the claim fails.

2.1step 1.1L2∎

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

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) for all same-sized square matrices