det2

In mathematics, det2 is a shorthand sometimes used to denote the determinant of a 2x2 matrix. For a matrix A = [[a, b], [c, d]], the det2(A) equals ad − bc.

The determinant of a 2x2 matrix represents the area-scaling factor of the linear transformation associated with A, and its sign encodes orientation. In two dimensions, det2(A) equals the signed area of the parallelogram spanned by the columns (or rows) of A.

Key properties: det2 is linear in each column, it changes sign if two columns are swapped, det2(A) = 0 if and only if the columns are linearly dependent, and det2(A) ≠ 0 if and only if A is invertible. When A is invertible, its inverse is A^{-1} = (1/det2(A)) [[d, -b], [-c, a]].

Example: det2([[1, 2], [3, 4]]) = 1·4 − 2·3 = −2.

Applications include solving 2x2 linear systems via Cramer's rule, computing area and orientation in geometry, and performing change of variables in double integrals where the Jacobian determinant is the det2 of the Jacobian matrix. The concept also connects to broader determinant theory and to the interpretation of linear maps in two dimensions.

See also: Determinant; 2x2 matrix; Cramer's rule; Jacobian.