2x2matrices

A 2x2 matrix is a square matrix with two rows and two columns, commonly written as A = [ [a, b], [c, d] ] with a, b, c, d from a field such as the real numbers. Key properties include the determinant det(A) = ad − bc and the trace tr(A) = a + d. The determinant indicates the area scaling of the associated linear transformation; det(A) = 0 means A is singular.

If det(A) ≠ 0, A is invertible, and its inverse is A^{-1} = (1/(ad − bc)) [ [d, −b], [−c, a] ]. The inverse satisfies AA^{-1} = A^{-1}A = I. The transpose is A^T = [ [a, c], [b, d] ], and (A^T)^{-1} = (A^{-1})^T.

Matrix operations: addition and subtraction are performed elementwise, while multiplication uses the dot-product rule. In general AB ≠ BA. Eigenvalues λ satisfy the characteristic equation λ^2 − tr(A)λ + det(A) = 0, linking trace and determinant to the spectrum of A.

Common forms include diagonal matrices with b = c = 0, and triangular matrices with one off-diagonal zero. Applications of 2x2 matrices include solving two-equation linear systems, representing 2D linear transformations, and, in computer graphics, performing rotations, scalings, and projections.