2×2Matrix

2×2Matrix refers to a 2-by-2 square matrix, a fundamental object in linear algebra used to represent linear transformations on two-dimensional space. It consists of four elements arranged in two rows and two columns: A = [ [a, b], [c, d] ].

Key properties include the determinant, det(A) = ad − bc, and the trace, tr(A) = a + d. The rank and invertibility depend on the determinant: A is invertible precisely when det(A) ≠ 0. In that case, the inverse is A⁻¹ = (1/(ad − bc)) [ [d, −b], [−c, a] ].

Multiplication defines the action of A on vectors in R². For a column vector x = [x₁; x₂], the product Ax yields a new vector, describing a linear transformation such as rotation, scaling, reflection, or shear. 2×2 matrices also form a 4-dimensional vector space under standard matrix addition and scalar multiplication, with a common basis given by the matrices E11, E12, E21, and E22.

Special cases help illustrate structure. The identity matrix I₂ = [ [1, 0], [0, 1] ] acts as the identity, while the zero matrix has all entries zero. Diagonal matrices diag(a, d) apply independent scaling along the coordinate axes. Rotations and reflections in the plane can be represented by specific 2×2 matrices, illustrating how A encodes geometric transformations.

Eigenvalues are roots of the characteristic equation λ² − tr(A)λ + det(A) = 0, with eigenvectors indicating invariant directions under the transformation A. 2×2Matrices are widely used to solve linear systems, analyze transformations, and study properties such as orientation preservation (det(A) > 0) or reversal (det(A) < 0).