Fix `mypy` errors in `lu_decomposition.py` (attempt 2) (#8100)

* updating DIRECTORY.md

* Fix mypy errors in lu_decomposition.py

* Replace for-loops with comprehensions

* Add explanation of LU decomposition and extra doctests

Add an explanation of LU decomposition with conditions for when an LU
decomposition exists

Add extra doctests to handle each of the possible conditions for when a
decomposition exists/doesn't exist

* updating DIRECTORY.md

* updating DIRECTORY.md

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