The minimal and characteristic polynomials agree iff there is a cyclic vector

Let \(k\) be a field and \(A \in \Mat_{n\times n}(k)\) with \(n \geq 1\), and let \(\min_A(x)\) be the minimal polynomial of \(A\). Then \(\min_A(x) = \det(xI - A)\) if and only if \(A\) admits a cyclic vector: some \(v \in k^n\) such that \(v, Av, A^2v, \ldots\) span \(k^n\).