Thoughts
I have decided WolframAlpha doesn't understand how to multiply matrices. Which is still concerning, but slightly less so.
Compare https://www.wolframalpha.com/input/?i=matrix+multiplication+calculator&assumption={"F"%2C+"MatricesOperations"%2C+"theMatrix2"}+->"{{1%3B0%3B0}%3B{0%3B1%3B1}%3B{0%3B1%3B2}}"&assumption={"F"%2C+"MatricesOperations"%2C+"theMatrix1"}+->"{{1%3B0%3B0}%3B{0%3B1%3B0}%3B{0%3B-1%3B1}}"
and https://www.wolframalpha.com/input/?i={{1%3B0%3B0}%3B{0%3B1%3B0}%3B{0%3B-1%3B1}}+*+{{1%3B0%3B0}%3B{0%3B1%3B1}%3B{0%3B1%3B2}}
Why are they different? I don't know. You don't know. No one knows.