Proof of Cayley-Hamilton

This short article presents a concise and elementary proof of the Cayley–Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial.Unlike many classical demonstrations relying on eigenvalues, canonical forms, or advanced algebraic theory, this proof uses only the properties of the adjugate matrix and basic polynomial identities. The exposition emphasizes clarity and accessibility, making it well-suited for undergraduate students or anyone seeking a direct, algebraic understanding of the theorem.All arguments are self-contained and presented in a compact and pedagogical format.