Let \(k\) be a field and \(A\in\Mat_{n\times n}(k)\). The characteristic polynomial of \(A\) is \[ \chi_A(x) \coloneqq \det(xI - A)\in k[x]. \]
\(\chi_A\) is monic of degree \(n\). Some texts define the characteristic polynomial as \(\det(A-xI)=(-1)^n\chi_A(x)\).
If \(\SNF(xI-A)=\diag(d_1,\ldots,d_n)\) is the Smith normal form of \(xI-A\) over \(k[x]\), with monic invariant factors \(d_i\), then \(\det(xI-A)\) and \(d_1\cdots d_n\) are monic and differ by a unit of \(k[x]\), hence are equal, and \[ \chi_A(x)=\det(xI-A)=\det\SNF(xI-A). \]