Finding the fundamental matrix for dummies

Good day, Habrahabr!
This post is for those who want to learn how to build the Fundamental Matrix of solutions of the System of Linear Differential Equations , but are afraid.

Background

If you google a little on this topic, then you can find a sufficient number of articles, most of which describe the construction of the Fundamental matrix through Jordan forms.
It seems to me that it is quite difficult to understand like this right away, and even more so to realize it.

History

Once I wanted to open a browser, drive into the search engine “Build a fundamental matrix easy / fast / understandable / NOT through Jordan forms” and get a ready-made algorithm, and so that it was, as they say, on a silver platter.

Algorithm

Here I give one interesting and easy to understand way of constructing the Fundamental matrix, which was once told to me by a wonderful teacher. This method is called "polynomial . " Perhaps someday it will come in handy for you. I expect the reader to be familiar with all the related definitions, such as “eigenvalue”, “qualifier”, etc.

A bit of theory

Fix in practice

You won’t be fed up with just one theory, so consider this method in action:

Conclusion

I hope that I expressed my thoughts quite clearly. And I apologize for my possible ignorance about the name of this method. If the method described is known under other names, I would like to know them. Thank you all for your attention!