Please help me with this math problem?

Question

Find a fundamental matrix for x'=(3, 2, -2, -2)x. (2x2 matrix, 3 and 2 on the left, -2 and -2 on the right.)

Answer 1

I answered this question yesterday, but it is no longer there. Here it is again:

 

The fundamental matrix of a system of homogeneous differential equations is the matrix whose columns are independent fundamental solutions of the system. Recall that each fundamental solution is of the form

 

x=s e^rt,

 

where r is an eigenvalue of the original matrix and s the corresponding eigenvector.

 

For your matrix, the eigenvalues are -1 and 2, with corresponding eigenvectors (1,2) and (2,1). Therefore, a fundamental solution set is

 

{ (1,2) e^-t , (2,1) e^2t }

 

Therefore, the fundamental matrix, using your notation, is

 

( e^-t, 2e^-t, 2e^2t , e^2t ).