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 is afraid.

Prehistory

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 this immediately, and even more so to realize.

Story

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

Algorithm

Here I present one interesting and easy to understand method of constructing the Fundamental Matrix, which was once told to me by a wonderful teacher. This method is called "polynomial" . Maybe someday it will be useful to you. I expect that the reader is familiar with all the accompanying definitions, such as "own number", "determinant", etc.
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Little theory

We fix in practice

One theory will not be full, so consider this method in action:

Conclusion

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