Characteristic Polynomial Calculator — 2×2 Matrix Eigenvalues

Formula

How this is calculated

The characteristic polynomial of a matrix A is defined as p(λ) = det(A − λI), where I is the identity matrix and λ is the eigenvalue parameter. Setting this equal to zero gives the eigenvalue equation. For a 2×2 matrix [[a, b], [c, d]], subtracting λ from each diagonal entry gives [[a−λ, b], [c, d−λ]], and its determinant expands to (a−λ)(d−λ) − bc = λ² − (a+d)λ + (ad−bc). The coefficient of λ is the negative trace and the constant term is the determinant, so the polynomial is always: λ² − tr(A)·λ + det(A) = 0.

Solving the quadratic with the quadratic formula gives eigenvalues λ = (tr ± √Δ) / 2 where Δ = tr² − 4·det is the discriminant. If Δ > 0 the matrix has two distinct real eigenvalues; if Δ = 0 there is one repeated real eigenvalue; if Δ < 0 the eigenvalues form a complex conjugate pair a ± bi where a = tr/2 and b = √(−Δ)/2. Eigenvalues encode important geometric properties: the trace equals their sum, the determinant equals their product, and the sign of the determinant reveals whether the transformation preserves or reverses orientation.

The grid animation shows the linear transformation defined by A: the unit square (dashed) is mapped to the shaded parallelogram, whose area equals |det(A)|. When det(A) > 0 orientation is preserved; when det(A) < 0 it is reversed (the parallelogram flips). This calculator covers only 2×2 matrices; for 3×3 and larger, the characteristic polynomial is a cubic or higher and requires more advanced solvers.