You misunderstood the question. The question is not about the general case. It is about solving a linear system of two equations and two variables. The solution is not found by "an algorithm", it is a simple formula that fits in one line.
Realistically speaking, though, if you're doing one 2x2 computation, you're probably going to end up doing a bunch of other linear algebra as well. If you don't do linear algebra in your everyday work, the threshold from rolling-your-own solution to grabbing a library is going to come up very quickly.
The Armstrong example is perhaps a bit mis-titled. He's talking about "implementing from scratch" only insofar as he can use the facilities of his platform to perform exactly the task he needs to do rather than bringing in a huge dependency, in this case, an FTP server that does a lot more than the one task at hand.
The question asked about inverting a 2x2 matrix, nothing about solving a linear system of equations. The latter is certainnly one common use of inverting a matrix, sure, but from the problem as given, the use-case could just as well be "report to the end user what the inverse is".
The invertibility issue also plagues any library and while it may not blow up by a division by zero we likely still would need to handle that case. This requires to understand the libraries handling of corner cases and how it affects the results.
You triggered a conversation thread that made me understand the ecosystem dynamics a bit better. Thank you.