TITLE: I need help with applied Math using homogeneous linear systems to solve ordinary differential equations.
QUESTION [6 upvotes]: Solve the IVP with initial values 
x(0)=(1,0,-2)
$$A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
So far My work consists of me finding the eigenvectors and eigenvalues first.
$$det(A-LI) = det(\begin{bmatrix} 1-L & 0 & 1 \\ 1 & 1-L & 0 \\ 0 & 0 & 1-L\end{bmatrix})=0$$
L = (1,1,1)
(0,0,1;1,0,0;0,0,0)(x;y;z)=(0;0;0)
$$A-1 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$$ * 
$$\begin{bmatrix} x\\y\\z\end{bmatrix}$$
(0;1;0)=v1
(0,0,1;1,0,0;0,0,0)(x1;y1;z1)=(0;1;0)
(0;0;1)=v2
(0,0,1;1,0,0;0,0,0)(x2;y2;z2)=(0;1;1)
(0;0;0)=v3
using the general solution I get 
x(t)=c1(0;1;0)e^t+c2((0;1;0)te^t+(0;0;1)e^t)+c3(t^2(0;1;0)e^t+t(0;0;1)e^t+(0;0;0)e^t)
Is this right so far? I can't find many online resources for 3 repeated eigenvalues.

REPLY [6 votes]: first find the eigen values. when you take the determinant you get solve for landa and get landa = 0,1,-1 and you use these values to find the eigen vectors when you row reduce