= Solution
Define
$$
P(A)=(-1)^{n-1}A^{n-1}
-c_{n-1}A^{n-2}-\cdots-c_2A-c_1I.
$$
The <Cayley-Hamilton theorem> for
$$
\chi_A(z)=(-1)^nz^n+c_{n-1}z^{n-1}
+\cdots+c_1z+c_0
$$
gives
$$
AP(A)=P(A)A=c_0I=\det(A)I.
$$
If $A$ is nonsingular, multiplication by $A^{-1}$ shows $P(A)=\det(A)A^{-1}=\operatorname{adj}(A)$.
For arbitrary $A$, apply the nonsingular result to $A-tI$ for a sequence of nonzero $t\to0$ avoiding the finitely many singular values. Both the characteristic coefficients and the adjugate entries depend polynomially on the matrix entries, so the limit gives
$$
\boxed{
\operatorname{adj}(A)
=(-1)^{n-1}A^{n-1}
-c_{n-1}A^{n-2}-\cdots-c_2A-c_1I}.
$$
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