2x2Matrix

2x2Matrix refers to a matrix with two rows and two columns. It is commonly denoted by A or M when described as A = [ [a, b], [c, d] ]. In mathematics, entries are taken from a field F (often the real numbers R or complex numbers C). The determinant is det(A) = ad − bc. If det(A) ≠ 0, A is invertible, and its inverse is A^{-1} = (1/(ad − bc)) [ [d, −b], [−c, a] ]. If det(A) = 0, A is singular and has no inverse.

The trace is tr(A) = a + d. The characteristic polynomial is λ^2 − tr(A) λ + det(A) = 0; eigenvalues λ = (tr(A) ± sqrt(tr(A)^2 − 4 det(A)))/2.

2x2 matrices represent linear transformations of the plane. They can perform scaling, rotation, reflection, shear, and combinations. Particular forms include diagonal matrices diag(a, d) and rotation matrices R(θ) = [ [cos θ, −sin θ], [sin θ, cos θ] ] with det 1. Multiplication of two 2x2 matrices corresponds to composition of their associated linear transformations.

Applications include solving systems of linear equations, computer graphics, and coordinate changes. The set of all invertible 2x2 matrices forms the group GL(2, F).

Example: Consider A = [ [2, 1], [0, 3] ]. Its determinant is 6, inverse is (1/6) [ [3, −1], [0, 2] ]. Its characteristic polynomial is λ^2 − 5λ + 6, so eigenvalues are 2 and 3.

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