Every square matrix over a commutative ring satisfies its own characteristic polynomial: if p(λ) = det(λI - A) is the characteristic polynomial of an n×n matrix A, then p(A) = 0 (the zero matrix), expressing a polynomial identity fulfilled by A.

Principle

Matrices obey their characteristic polynomial because the polynomial relation captures the algebraic dependencies among powers of the linear operator; this yields a reduction of high powers to lower-degree polynomial expressions in A.

Demonstration

For a 2×2 matrix A = [[a,b],[c,d]] the characteristic polynomial p(λ) = λ^2 - (a+d)λ + (ad-bc) and direct substitution shows p(A) equals the zero matrix; concretely, A^2 - (trace A)A + (det A)I = 0.

Misapplication

Applying Cayley–Hamilton to non-square matrices or treating the theorem as giving independent minimal-polynomial factors is incorrect; also naive application over noncommutative coefficient rings requires care about order of multiplication.

Consequence

Cayley–Hamilton implies that any polynomial in A reduces modulo p to a polynomial of degree < n, enabling computation of matrix functions, derivation of explicit formulas for powers, and relations with the minimal polynomial.

Reversal

The converse is not generally true: satisfying a polynomial does not uniquely determine the characteristic polynomial, since many polynomials may annihilate A; the minimal polynomial divides the characteristic polynomial but can be strictly smaller.

Boundary

The classical statement assumes square matrices over a commutative ring (commonly a field); for operators on infinite-dimensional spaces or matrices over noncommutative rings, formulations and validity require additional hypotheses.

Semantic Tension

There is tension between Cayley–Hamilton as an existence identity (p(A)=0) and the separate concept of the minimal polynomial which gives the smallest annihilating polynomial; distinguishing these clarifies spectral and algebraic structure.

Synthesis

Cayley–Hamilton provides a concrete algebraic relation for linear operators: the characteristic polynomial annihilates the matrix, which compresses algebraic information about A into a usable polynomial identity for computations and structural analysis.