Systems of higher order equations P(z)y = 0. The equivalence of polynomial matrices.

Let us see by way of an example how we can solve a system of higher order difference equations. Consider 14.1 $$\begin{array}{*{20}{c}} {{{y}_{2}}^{{\prime \prime }} + {{y}_{1}}^{\prime } + {{y}_{2}}^{\prime } + {{y}_{3}}^{\prime } - {{y}_{1}} - 3{{y}_{2}} - {{y}_{3}} = 0} \hfill \\ {{{y}_{1}}^{{\prime \prime \prime }} - {{y}_{1}}^{{\prime \prime }} + {{y}_{3}}^{{\prime \prime }} - 4{{y}_{1}}^{\prime } - 4{{y}_{3}}^{\prime } + 2{{y}_{1}} = 0} \hfill \\ {{{y}_{1}}^{\prime } + {{y}_{2}}^{\prime } - {{y}_{1}} - {{y}_{3}} = 0.} \hfill \\ \end{array}$$