The statement that every square matrix satisfies its own characteristic polynomial: if p(λ) = det(λI − A) is the characteristic polynomial of a square matrix A, then p(A) = 0 (the zero matrix).

Principle

A polynomial constructed from the determinant captures algebraic relations of a matrix; substituting the matrix into that polynomial yields an annihilating polynomial, so the characteristic polynomial annihilates the matrix.

Demonstration

For a 2×2 matrix A = [[a, b], [c, d]], the characteristic polynomial is λ^2 − (a + d)λ + (ad − bc). Replacing λ by A gives A^2 − (trace A) A + (det A) I = 0, an explicit matrix identity.

Misapplication

Applying the theorem to non-square matrices or to square matrices without attention to the base ring (for example assuming it holds verbatim over rings where determinants are not well-behaved) leads to incorrect conclusions.

Consequence

Ensures the characteristic polynomial is an annihilating polynomial and implies the minimal polynomial divides the characteristic polynomial; it underpins computations of matrix functions, derivation of identities involving trace and determinant, and structural decompositions.

Reversal

The converse — that every annihilating polynomial must be a multiple of the characteristic polynomial — is false in that not every annihilating polynomial equals the characteristic polynomial, but the minimal polynomial always divides the characteristic polynomial.

Boundary

Holds for square matrices over commutative rings or fields where the characteristic polynomial is defined via determinant; caution is needed over noncommutative coefficient rings or in contexts lacking a determinant theory.

Semantic Tension

Competes with the viewpoint of the minimal polynomial and Jordan canonical form: Cayley–Hamilton gives a universal annihilator (the characteristic polynomial), while the minimal polynomial captures the smallest annihilating relation and more refined structure.

Synthesis

Cayley–Hamilton unites determinant-based spectral data with algebraic relations of matrices: the characteristic polynomial, computed from trace and determinant data, when evaluated at the matrix, yields the zero operator and links eigenvalue information to polynomial functional calculus.