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« on: November 21, 2013, 12:21:49 PM »
\begin{array}{l}\det (A - rI) = \left( {\begin{array}{*{20}{c}}{3 - r}&{ - 2}\\2&{ - 2 - r}\end{array}} \right)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {r^2} - 1 - 2 = 0\\r = 2,\;r = - 1\\{\rm{when}}\;r = 2\\1{\xi _1} = 2{\xi _2}{\rm{and}}{\xi ^1} = \left( {\begin{array}{*{20}{c}}2\\1\end{array}} \right)\\{\rm{when}}\;r = - 1\\2{\xi _1} = 1{\xi _2}{\rm{and}}{\xi ^2} = \left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right)\\x = {c_1}\left( {\begin{array}{*{20}{c}}2\\1\end{array}} \right){e^{2t}} + {c_2}\left( {\begin{array}{*{20}{c}}1\\2\end{array}} \right){e^{ - t}}\end{array}
\begin{array}{l}{\rm{for }}{c_1} = 0,as\;t \to + \infty ,x \to 0.as\;t \to - \infty ,x \to + \infty \\{\rm{for }}{c_1} \ne 0,as\;t \to + \infty ,{\rm{the first term dominates, so }}x \to + \infty .\\{\rm{ }}as\;t \to - \infty ,{\rm{the second term dominates}},sox \to + \infty \end{array}