2x2

Two by two, often written as 2x2, is a shorthand for a two-row by two-column arrangement or for dimensions described as two units by two units. The term is used across mathematics, statistics, and grid-based contexts to denote a small, square structure.

In mathematics, a 2x2 matrix is a square array with two rows and two columns. It is typically written as [[a, b], [c, d]] and is used to represent linear transformations of two-dimensional vectors. Key concepts include the determinant ad − bc, which determines invertibility, and if the determinant is nonzero, the matrix has an inverse given by (1/det) times a swapped and sign-changed version of the original. The identity matrix for 2x2 is [[1, 0], [0, 1]]. Two-by-two matrices describe linear transformations in the plane; when translations are involved, a 3x3 matrix in homogeneous coordinates is used instead.

In statistics, a 2x2 contingency table summarizes the frequency of observations across two categorical variables, with four cells containing counts. It is a basic tool for chi-square tests of independence and, with small sample sizes, for Fisher’s exact test.

In geometry and combinatorics, a 2x2 grid partitions a area into four equal squares, yielding a unit area of 4 if the unit square has side length 1. Such grids underpin tiling, counting problems, and simple spatial reasoning.

In everyday language, “two by two” also denotes pairing or grouping in twos, and is famously used in various historical or cultural contexts to describe traveling or gathering in pairs.