The Formula of the Elementary Solution Matrix of the 3-Dimensional Linear System

It is well known that the solution of the n-dimensional constant coefficient linearsystem X=AX, X(0)=η can be represented as  in which, for all j, λ_i, is the n-th characteristic root of A, and v_j is the element ofthe solution space of (A-λ_j I)ⁿ ju=0 and η=v₁ +… +v_k. From this, we can getthe elementary solution matrix EXPAt of the system. However, this is not a conve-nient method if it is used directly in computation, because to find v_j would involve acomplicated process. This paper gives a direct calculation method where by to solve theproblem of the 3-dimensional systems. The EXPAt can be represented by the elemen-ts of A, which will bring about convenient in calculation.