Modification methods of building minimum polynomials

In this article is leading modification method of building minimum polynomial with help solution systems of linear algebraic equations. The proposed approach, isn't changes base idea of method is gives possibility in higher degree is reduces number of calculations. If early for building coefficients of minimum polynomial of matrix $n$ - order by employment the method sufficiently was researched the solution of systems linear algebraically equations order to $n^2\times m$, $n<m$, then in modification method for this sufficiently is researches solution of systems linear algebraically equations in order to $n\times m$, $n<m$.