Determinants of real 2×2 matrices

Mathematical statement

For M=[[a,b],[c,d]] over R, define D(M)=det(M)=ad−bc. M is invertible over R exactly when det(M)≠0. For 2×2 real matrices A,B, det(AB)=det(A)det(B), and det(tM)=t²det(M).

Representative formula (exact TeX)

D(M)

Assumptions

• The matrices are 2×2 and the field is R. Do not transfer the real-field nonzero-determinant criterion unchanged to the integer ring.
• Integer entries and integer t form the exact exercise slice. A negative t acts through t², and t=0 gives determinant zero for the scaled matrix.
• AB is the usual row-by-column matrix product, not an entrywise product. The determinant product identity does not assert that A and B commute.

Symbol scope

• D — scoped 2×2 determinant operation
• M — 2×2 real square matrix

Intuition, definition, and misconceptions

Signs and positions matter. Do not replace ad−bc by ad+bc or take an absolute value of a negative determinant. A nonzero determinant may be positive or negative and need not equal one.

Invertibility over R is the exact condition for this slice. Having integer entries does not imply that an inverse also has integer entries or is an inverse over the integer ring.

Acceptance recomputes determinants of the actual AB and tM, while the producer may use the product and t² identities as a different algorithm. These finite checks add no Evidence for higher-order determinant theory.

Common errors include requiring a nonzero determinant to be positive, interpreting AB entrywise, or omitting the field from an invertibility criterion.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite matrices: dimensions, indexing, and transpose — First establish square dimensions and entry positions.
• Real number — The invertibility criterion explicitly uses the real field.

Verification boundary

Integer 2×2 determinant instances can be recomputed; general dimensions, determinant theory, and field/ring bridges await formalization.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bridge pinned det_fin_two, det_mul, det_smul, and nonsingular inverse APIs over the real field.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.