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

[tianyizheng02]

Describe your change:

• Fixed type hints in lu_decomposition.py to resolve mypy errors (see #8070)
• Added explanation of LU decomposition
• Added doctests to handle different cases in which an LU decomposition does or doesn't exist

Attempt 2 of making a PR since the first one (#8076) was closed due to having too many PRs open at a time.

• [ ] Add an algorithm?
• [x] Fix a bug or typo in an existing algorithm?
• [x] Documentation change?

Checklist:

• [x] I have read CONTRIBUTING.md.
• [x] This pull request is all my own work -- I have not plagiarized.
• [x] I know that pull requests will not be merged if they fail the automated tests.
• [x] This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• [ ] All new Python files are placed inside an existing directory.
• [ ] All filenames are in all lowercase characters with no spaces or dashes.
• [x] All functions and variable names follow Python naming conventions.
• [x] All function parameters and return values are annotated with Python type hints.
• [x] All functions have doctests that pass the automated testing.
• [x] All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• [x] If this pull request resolves one or more open issues then the commit message contains Fixes: #{$ISSUE_NO}.

[tianyizheng02]

@cclauss mind taking a look?

[tianyizheng02]

@cclauss Not sure why this still has the "tests are failing" label since it looks like all checks passed. Either way, does this look good to be merged?