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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.)

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Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
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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 ert,
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) e2t }
Therefore, the fundamental matrix, using your notation, is
( e-t, 2e-t, 2e2t , e2t ).


The answer is ( e-t, 2e-t, 2e2t , e2t ). This is a 2x2 matrix, e-t and 2e-t is on the left, 2e2t and e2t on the right.