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Re: the first sin, I was told to not invert matrices but never given a satisfying reason. So I wrote a blog post about it:

http://gregorygundersen.com/blog/2020/12/09/matrix-inversion...

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The other obvious reason to avoid inverses is that they're only defined for square matrices, whereas LU decomposition works on general rectangular matrices (or rather PLU decomposition,same basic idea).

Invert can also destroy structure you might want to keep around, for example the LU factorization of a banded matrix will still be banded (lapack will do partial pivoting which will increase the bandwidth, but with it'll only double the number of super-diagonals in L), while the inverse is a full matrix.

Err, the number of super-diagonals in U obviously, haha, oops.

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