On the characteristic polynomial of matrices with prescribed rows (Q1963928)

Two matrices \(A\in F^{n \times n}\) and \(B\in F^{n \times t}\), where \(F\) is an arbitrary field, are given. The characteristic polynomial of the product \([A B]\) is defined as the product of the invariant factors of \([xI_n - A - B]\). Possible characteristic polynomials of \([A B]\), when some of its rows are prescribed and remaining rows vary, are deduced. This is a generalization of the same case with \(t=0\) solved by \textit{H. K. Wimmer} [Monatsh. Math. 78, 256-263 (1974; Zbl 0288.15014)].