In this talk, we discuss the combinatorial problem on the number of matrices with coefficients in a finite field having a given characteristic polynomial. This problem has been studied in three different paper in the late 1960s. We will see one of its proof technique and move to the next goal of counting the number of m-companion (multi-companion) matrices with given characteristics polynomial.

We will see that this problem has a connection with {\sigma}-splitting subspace problem, which was posed by Niederreiter (1995) and on the number of LFSR with multi-input and multi-output delay elements. Lastly, we will state some open problems related to this (hoping we will solve it together). This talk will be self contianed.