2x2s

2x2 matrices, commonly abbreviated as 2x2s, are square arrays with two rows and two columns. They are typically written as [ [a, b], [c, d] ]. In linear algebra, a 2x2 matrix represents a linear transformation from R^2 to R^2, defined by how it maps the standard basis vectors.

Operations on 2x2s include addition, subtraction, and multiplication, as well as scalar multiplication. The determinant det = ad − bc is a scalar that encodes area scaling and invertibility: a 2x2 matrix is invertible if det ≠ 0, and its inverse is (1/det) [ [d, -b], [-c, a] ]. The trace, a + d, is another characteristic of the matrix. Eigenvalues and eigenvectors can be found for 2x2 matrices and provide insight into the transformation’s behavior under repeated application.

Special forms and interpretation help intuition. The identity matrix I2 = [ [1, 0], [0, 1] ] leaves every vector unchanged. Rotation matrices, such as R(θ) = [ [cos θ, -sin θ], [sin θ, cos θ] ], rotate vectors by θ in the plane. Diagonal matrices like diag(s1, s2) = [ [s1, 0], [0, s2] ] perform independent scaling along axes. In general, the columns of a 2x2 matrix are the images of the standard basis vectors, so the matrix encodes how the transformation reshapes the plane.

Applications of 2x2 matrices are broad. They are used to solve systems of two linear equations, to perform 2D computer graphics transformations (rotation, scaling, shearing), and in physics and engineering models that involve linear transformations in two dimensions. They provide a compact, computationally efficient framework for understanding and applying linear mappings.