Characteristic polynomial of a matrix

Let \(k\) be a commutative ring and \(A\) an \(n\times n\) matrix over \(k\). The characteristic polynomial of \(A\) is \[ p_A(x) \coloneqq \det(xI_n - A)\in k[x], \] where \(I_n\) is the \(n\times n\) identity matrix.