2×2matrisen

A 2×2 matrix is a matrix with two rows and two columns. It is typically written as A = [[a, b], [c, d]], where a, b, c, d come from a field such as the real or complex numbers. Real 2×2 matrices act as linear transformations on the two-dimensional vector space, sending pairs (x, y) to another pair (ax + by, cx + dy).

The determinant of a 2×2 matrix is det(A) = ad − bc. The determinant measures area scaling under the associated linear transformation and indicates orientation: a positive determinant preserves orientation, a negative determinant reverses it, and a determinant of zero means the transformation collapses the plane to a line or a point.

A matrix is invertible exactly when det(A) ≠ 0. In that case, its inverse is A⁻¹ = (1/(ad − bc)) [[d, −b], [−c, a]]. The trace, tr(A) = a + d, is the sum of the diagonal entries and appears in many formulas, including the characteristic polynomial.

2×2 matrices form a closed set under multiplication; the product corresponds to the composition of the associated linear maps. The eigenvalues of A are the roots of the characteristic polynomial p(λ) = λ² − (tr A) λ + det A. By the Cayley–Hamilton theorem, A satisfies its characteristic equation: A² − (tr A) A + (det A) I = 0.

Special forms include diagonal matrices diag(λ1, λ2), which represent independent scaling along coordinate axes, and rotation or shear matrices, which describe rotations and distortions in the plane. Diagonalization is possible when A has two linearly independent eigenvectors; otherwise, one may work with Jordan forms.

2×2 matrices are fundamental tools in solving linear systems, computer graphics, differential equations, and many areas of applied mathematics due to their simplicity and rich structure.