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Why is the following not possible:

$$ b=(X'X)^{-1}X'y = X^{-1}(X')^{-1}X'y=X^{-1}y $$

While this term $(AB)^{-1}=B^{-1}A^{-1}$ applies to any two matrices as long as both are of full rank and are $nxn$, the problematic part is when X is not $nxn$. $X'X$ always is a nxn matrix, and hence may have an inverse. Non squared matrices need not have an inverse.

Any other reason to add?

cascom
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    The $X$ matrix is never $n \times n$ except in near-pathological cases. – Forgottenscience Sep 27 '20 at 18:55
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    The duplicate was found in the list of "Related" threads at the bottom right: it's a good thing to consult before posting a question, because often it shows you the answer right away. – whuber Sep 27 '20 at 19:21

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