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ماتریس زیر را در نظر بگیرید

\begin{align*}
P(\lambda ) &= \det (\lambda I - {J_0})\\
& = \left| {\begin{array}{*{20}{c}}
{\lambda  + (\nu  + {u_1} + {\lambda _4} - \nu {p_2})}&{{\lambda _4} - \nu {p_2}}&{\rho {S_0} + {\lambda _4} - \nu {p_2}}&{{\lambda _4} + \nu {p_1} - \nu {p_2} + \theta \rho {S_0}}\\
0&{\lambda  + (\nu  + {\lambda _1})}&{ - \rho {S_0}}&{ - \rho \theta {S_0}}\\
0&{ - {\lambda _1}}&{\lambda  + (\nu  + {\lambda _2})}&0\\
0&0&{ - {p_3}{\lambda _2}}&{\lambda  + (\nu  + {\lambda _3} + {u_2} - \nu {p_1})}
\end{array}} \right|\\
P(\lambda ) &= (\lambda  + {l_0})({\lambda ^3} + {l_1}{\lambda ^2} + {l_2}\lambda  + {l_3})
\end{align*}

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