The statement that every square matrix over a commutative ring (in particular over a field) satisfies its own characteristic polynomial: if p(λ) = det(λI − A) then p(A)=0 when p is evaluated with matrix substitution.

Principle

Substituting the matrix into its characteristic polynomial yields the zero matrix because the polynomial annihilates the linear transformation represented by the matrix.

Demonstration

For a 2×2 matrix A = [[a,b],[c,d]], its characteristic polynomial p(λ)=λ^2−(a+d)λ+(ad−bc); Cayley–Hamilton asserts p(A)=A^2−(a+d)A+(ad−bc)I = 0, an explicit matrix identity that can be checked by calculation.

Misapplication

Plugging A into a polynomial with coefficients that do not commute with A (noncommutative coefficient ring) without checking centrality, or confusing evaluation of scalar polynomials at A with pointwise substitution of scalars.

Consequence

Allows expressing higher powers of A as lower-degree polynomials in A, provides a route to compute matrix functions and inverses (when invertible) by polynomial reduction, and links characteristic and minimal polynomials.

Reversal

The converse is not automatic: a polynomial q with q(A)=0 need not equal the characteristic polynomial; it must divide the minimal polynomial, which in turn divides the characteristic polynomial.

Boundary

Standard Cayley–Hamilton holds for matrices over commutative rings and fields; extensions to matrices over noncommutative rings require care about the order of multiplication and coefficient centrality. It presupposes a well-defined characteristic polynomial.

Semantic Tension

Tension exists between the characteristic polynomial and the minimal polynomial: Cayley–Hamilton guarantees annihilation by the characteristic polynomial, but the minimal polynomial is the smallest annihilating polynomial and is often the sharper invariant for dynamics.

Synthesis

Cayley–Hamilton unifies linear algebraic structure by asserting that the characteristic polynomial annihilates its matrix, enabling algebraic reduction of matrix powers and providing a bridge between determinant-based invariants and operator-theoretic annihilators.