## Compute the fundamental matrix of a given linear system of first-order differential equations

Let and be matrices as defined in this previous exercise, with . Compute the fundamental matrix of this differential equation where is one of the following matrices:

Let be a matrix, and let be in Jordan canonical form. Using this previous exercise, we have that is a fundamental matrix, and by this previous exercise, is a fundamental matrix of our differential equation. We can use this previous result (and part 2 of this previous exercise) to compute .

We computed the Jordan canonical forms (and conjugators) for these three matrices in this previous exercise.

For the matrix , we have and . Now , and . We can easily verify that this is indeed a fundamental matrix.

For the matrix , as computed in this previous exercise, if we let then is in Jordan canonical form. Now . So is a fundamental matrix.

Finally, consider . Letting (Seethis previous exercise), we have in Jordan canonical form. Then , and hence is a fundamental matrix.

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