Determinant of a companion matrix

Expansion along the first column. The only nonzero entry in column $1$ is $C_{31}=-c_0$. Its cofactor is
$$(-1)^{3+1}\begin{vmatrix}1&0\\0&1\end{vmatrix}=+1.$$
Hence
$$\det C=(-c_0)\cdot(+1)=-c_0.$$

Remark. This is the companion matrix of $P(X)=X^3+c_2X^2+c_1X+c_0$. Its characteristic polynomial is
$$\det(\lambda I-C)=\lambda^3+c_2\lambda^2+c_1\lambda+c_0=P(\lambda),$$
which is monic. The constant term of the characteristic polynomial is $c_0$, and for an $n\times n$ companion matrix the determinant equals $(-1)^n$ times that constant term, so here $\det C=(-1)^3c_0=-c_0$, consistent with the direct computation.