Characteristic Polynomial Calculator

The characteristic polynomial of a square matrix A is defined as p(λ) = det(A − λI), where I is the identity matrix of the same size and det denotes the determinant. It is a polynomial in λ whose degree equals the size of the matrix. The roots of this polynomial are the eigenvalues of A.

For a 2×2 matrix [[a, b], [c, d]], subtract λ from each diagonal entry to get [[a−λ, b], [c, d−λ]], then take the determinant: p(λ) = (a−λ)(d−λ) − bc = λ² − (a+d)λ + (ad−bc). This simplifies to λ² − trace(A)·λ + det(A).

For a 3×3 matrix, form the matrix (A − λI) by subtracting λ from each diagonal entry, then compute its determinant using cofactor expansion. The result is a cubic polynomial: p(λ) = −λ³ + trace(A)λ² − (sum of 2×2 principal minors)λ + det(A). This calculator handles the expansion automatically.

The eigenvalues of a matrix are exactly the roots (zeros) of its characteristic polynomial p(λ). Setting p(λ) = 0 and solving for λ gives you all eigenvalues. For example, a 2×2 matrix has at most 2 eigenvalues, and a 3×3 matrix has at most 3 (counting multiplicity).

When you compute det(A − λI), expanding the determinant always produces a term (−λ)ⁿ from the diagonal entries, making the polynomial degree exactly n for an n×n matrix. All other terms have degree less than n, so the leading term always determines the degree.

For a 2×2 matrix, the coefficient of λ is −trace(A) and the constant term is det(A). In general, the coefficients are related to elementary symmetric polynomials of the eigenvalues. The constant term is always det(A) (up to sign) and the coefficient of λⁿ⁻¹ is always −trace(A).

Not always. The minimal polynomial of a matrix divides the characteristic polynomial and has the same roots, but may have lower degree. For example, the identity matrix has characteristic polynomial (λ−1)ⁿ but minimal polynomial (λ−1). The minimal polynomial is the smallest-degree polynomial that the matrix satisfies.

If your matrix entries are all real numbers, the characteristic polynomial will always have real coefficients. However, the roots (eigenvalues) may be complex. If matrix entries are complex, the polynomial coefficients will also generally be complex.