Cayley-Hamilton theorem

Let \[ p(t)=\det(tI-A)=c_0+c_1t+\cdots+c_nt^n. \] Over the polynomial ring \(R[t]\), the adjugate identity gives \[ (tI-A)\operatorname{adj}(tI-A)=p(t)I. \] Write \[ \operatorname{adj}(tI-A)=B_0+B_1t+\cdots+B_{n-1}t^{n-1}, \qquad B_i\in M_n(R). \] Comparing coefficients of powers of \(t\) yields \[ -AB_0=c_0I, \] \[ B_{k-1}-AB_k=c_kI\qquad(1\le k\le n-1), \] and \[ B_{n-1}=c_nI. \] Multiply these equations respectively by \[ I,A,A^2,\ldots,A^{n-1},A^n \] on the left and add them. The terms involving the \(B_i\) telescope, leaving \[ 0=c_0I+c_1A+\cdots+c_nA^n=p(A). \] Therefore \[ \boxed{p(A)=0}, \] which is the Cayley–Hamilton theorem.