feat: Add LU decomposition algorithm for matrices using Doolittle's algorithm

[Raghu0703]

Description

This PR implements the LU decomposition algorithm for square matrices as requested in issue #6833.

Changes

• Implements LU decomposition using Doolittle's method
• Decomposes square matrix A into L (lower) and U (upper) triangular matrices
• Includes comprehensive test cases with edge case handling
• Time complexity: O(n³), Space complexity: O(n²)
• Resolves [FEATURE REQUEST] Add LU decomposition algorithm for matrix (java) (hactoberfest'25) #6833

Algorithm Details

• Purpose: Decomposes a square matrix A into L × U where:
  • L is a lower triangular matrix with 1s on diagonal
  • U is an upper triangular matrix
• Use cases: Solving linear equations, computing determinants, finding matrix inverses
• Method: Doolittle's algorithm
• Time Complexity: O(n³)
• Space Complexity: O(n²)

Testing

• ✅ Basic 3x3 matrix decomposition (from issue example)
• ✅ Identity matrix decomposition
• ✅ 2x2 matrix decomposition
• ✅ Non-square matrix error handling
• ✅ Empty matrix error handling
• ✅ Singular matrix error handling

Checklist

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized it.
• All filenames are in PascalCase.
• All functions and variable names follow Java naming conventions.
• All new algorithms have a URL in their comments that points to Wikipedia or other similar explanations.
• All new code is formatted with clang-format -i --style=file path/to/your/file.java

Closes #6833

[codecov-commenter]

Codecov Report

❌ Patch coverage is 50.00000% with 27 lines in your changes missing coverage. Please review.
✅ Project coverage is 78.45%. Comparing base (f693c44) to head (aa709b7).

Files with missing lines	Patch %	Lines
...java/com/thealgorithms/matrix/LUDecomposition.java	50.00%	27 Missing ⚠️

Additional details and impacted files

@@             Coverage Diff              @@
##             master    #7101      +/-   ##
============================================
- Coverage     78.50%   78.45%   -0.06%     
- Complexity     6752     6757       +5     
============================================
  Files           759      759              
  Lines         22402    22432      +30     
  Branches       4400     4407       +7     
============================================
+ Hits          17587    17598      +11     
- Misses         4109     4129      +20     
+ Partials        706      705       -1     

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[github-actions]

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