Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 6A / d / Solution

DefineThe Cayley-Hamilton theorem forgivesIf $A$ is nonsingular, multiplication by $A^{-1}$ shows $P(A)=\det (A)A^{-1}=\mathrm{adj}(A)$.

For arbitrary $A$, apply the nonsingular result to $A-tI$ for a sequence of nonzero $t\to 0$ avoiding the finitely many singular values. Both the characteristic coefficients and the adjugate entries depend polynomially on the matrix entries, so the limit gives