Particular Solution to a System of First Order Linear Equations

[lionsgirl12]

Consider a certain linear system of 2 equations x'=Ax. Suppose the real-valued coefficient matrix A has r=-1+4i as one of its eigenvalues, and that one of its corresponding eigenvectors is (2, 5-2i).

Find the particular solution that satisfies the initial condition x(0) = (-4, 3).
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My work:

A is a 2x2 matrix with given initial condition:

x' = (a b, c d)
x(0) = (-4, 3)

r²-(a+d)-(ad-bc) = 0

Since r = -1+4i we know that lambda = -1 and mu = 4

Given k = (2, 5-2i) we can work backwards and we end up with (-5-2i, 2) which can be written as -5-2ix₁+2x₂=0

I am not sure how to proceed if we do not know the values for a b c and d.

 

[vela]

You should be able to say what the other root is since A is a real-valued matrix.