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

1.1

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.

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 · 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