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Initial Value Problem, Ordinary Differential Equation: Jordan and Diagonalization

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Find the solution of the system X' using the "diagonalization" technique (actually the Jordan form in this case)
Please see the attached file.

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Solution Summary

The solution to a system expressed as matrices is found. Initial value problems and ordinary differential equations are solved by using Jordan and Diagonalization.

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because,
X' = AX + g
A = = {(a11,a12,a13),(a21,a22,a23),(a31,a32,a33)}
=> A = {(2,0,0),(-1,0,-1),(1,1,2)}
Now,
(A - lI)X = 0
=> determinant (A-lI) = 0
=> l = 1,1,2 (= eigen values) => non diagonalizable
=> eigen vectors = (0,1,-1) for l =1, (1,-1,1) for l = 2 but for ...

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