= Solution
A finite <basis> is a generating set, so the preceding <determinant trick> applies with $R=k$ and $M=V$. If $p(t)=\det(A-tI)$, the map $k[t]\to k[\phi]$ sending $t$ to $\phi$ is a ring homomorphism. Since a <determinant> is a polynomial expression in its entries, it sends $p(t)$ to $\det(A-\phi I)=0$. Thus
$$
\boxed{p(\phi)=0.}
$$
With the more usual monic <characteristic polynomial> $\chi_A(t)=\det(tI-A)=(-1)^{\dim V}p(t)$, this reads $\chi_A(\phi)=0$, the <Cayley-Hamilton theorem>. If the matrix convention uses column coordinates instead of the row convention above, transpose $A$; its <characteristic polynomial> is unchanged.
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