Added rank of matrix in linear algebra (#8687)

* Added rank of matrix in linear algebra

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* Corrected name of function

* Corrected Rank_of_Matrix.py

* Completed rank_of_matrix.py

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* delete to rename Rank_of_Matrix.py

* created rank_of_matrix

* added more doctests in rank_of_matrix.py

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* fixed some issues in rank_of_matrix.py

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* added moreeee doctestsss in rank_of_mtrix.py and fixed some bugss

* Update linear_algebra/src/rank_of_matrix.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update linear_algebra/src/rank_of_matrix.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update linear_algebra/src/rank_of_matrix.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update rank_of_matrix.py

* Update linear_algebra/src/rank_of_matrix.py

Co-authored-by: Caeden Perelli-Harris <caedenperelliharris@gmail.com>

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: Caeden Perelli-Harris <caedenperelliharris@gmail.com>

linear_algebra/src/rank_of_matrix.py

@@ -0,0 +1,89 @@
+"""
+Calculate the rank of a matrix.
+
+See: https://en.wikipedia.org/wiki/Rank_(linear_algebra)
+"""
+
+
+def rank_of_matrix(matrix: list[list[int | float]]) -> int:
+    """
+    Finds the rank of a matrix.
+    Args:
+        matrix: The matrix as a list of lists.
+    Returns:
+        The rank of the matrix.
+    Example:
+    >>> matrix1 = [[1, 2, 3],
+    ...            [4, 5, 6],
+    ...            [7, 8, 9]]
+    >>> rank_of_matrix(matrix1)
+    2
+    >>> matrix2 = [[1, 0, 0],
+    ...            [0, 1, 0],
+    ...            [0, 0, 0]]
+    >>> rank_of_matrix(matrix2)
+    2
+    >>> matrix3 = [[1, 2, 3, 4],
+    ...            [5, 6, 7, 8],
+    ...            [9, 10, 11, 12]]
+    >>> rank_of_matrix(matrix3)
+    2
+    >>> rank_of_matrix([[2,3,-1,-1],
+    ...                [1,-1,-2,4],
+    ...                [3,1,3,-2],
+    ...                [6,3,0,-7]])
+    4
+    >>> rank_of_matrix([[2,1,-3,-6],
+    ...                [3,-3,1,2],
+    ...                [1,1,1,2]])
+    3
+    >>> rank_of_matrix([[2,-1,0],
+    ...                [1,3,4],
+    ...                [4,1,-3]])
+    3
+    >>> rank_of_matrix([[3,2,1],
+    ...                [-6,-4,-2]])
+    1
+    >>> rank_of_matrix([[],[]])
+    0
+    >>> rank_of_matrix([[1]])
+    1
+    >>> rank_of_matrix([[]])
+    0
+    """
+
+    rows = len(matrix)
+    columns = len(matrix[0])
+    rank = min(rows, columns)
+
+    for row in range(rank):
+        # Check if diagonal element is not zero
+        if matrix[row][row] != 0:
+            # Eliminate all the elements below the diagonal
+            for col in range(row + 1, rows):
+                multiplier = matrix[col][row] / matrix[row][row]
+                for i in range(row, columns):
+                    matrix[col][i] -= multiplier * matrix[row][i]
+        else:
+            # Find a non-zero diagonal element to swap rows
+            reduce = True
+            for i in range(row + 1, rows):
+                if matrix[i][row] != 0:
+                    matrix[row], matrix[i] = matrix[i], matrix[row]
+                    reduce = False
+                    break
+            if reduce:
+                rank -= 1
+                for i in range(rows):
+                    matrix[i][row] = matrix[i][rank]
+
+            # Reduce the row pointer by one to stay on the same row
+            row -= 1
+
+    return rank
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()