The largest invariant factor is the minimal polynomial.Suggest a slogan
Let \(k\) be a field, let \(A\in\Mat_n(k)\), and let \(f_1\divides f_2\divides\cdots\divides f_n\) be the invariant factors of \(A\): the monic diagonal entries of the Smith normal form of \(xI-A\) over \(k[x]\). Then the minimal polynomial of \(A\) is \[ \min_A(x)=f_n(x). \]
With \(x\) acting as \(A\), the \(k[x]\)-module \(k^n\) is isomorphic to \(\bigoplus_{j=1}^nk[x]/(f_j)\). A polynomial \(g\) satisfies \(g(A)=0\) if and only if \(g\) annihilates this module, that is, \(f_j\divides g\) for every \(j\); since \(f_j\divides f_n\) for all \(j\), this holds if and only if \(f_n\divides g\). So \(f_n\) is the monic generator of the ideal of polynomials vanishing at \(A\).