How to Convert a Given Matrix to a Symmetric Matrix with Minimum Row/Column Insert Operations in Python?

Question

I'm working on a programming challenge where I need to convert a given matrix to a symmetric matrix. The catch is that I need to accomplish this with the minimum number of row or column insertions. I can either insert a row or a column to make the matrix symmetric.

For example, given the following 3x3 matrix:

[
  [1, 2, 3],
  [4, 5, 6],
  [3, 6, 7]
]

One possible way to make it symmetric could be to insert a column to match the first and last rows, resulting in a symmetric 4x4 matrix, like so:

[
  [1, 2, 3, 1],
  [4, 5, 6, 4],
  [3, 6, 7, 3],
  [1, 4, 3, 1]
]

My main concern is to do this as efficiently as possible, both in terms of time and space complexity. Any suggestions or pointers would be much appreciated. I'm particularly interested in Python solutions.

My code is:

def make_symmetric(matrix):
    # Helper function: Check if the matrix is symmetric
    def is_symmetric(matrix):
        rows = len(matrix)
        cols = len(matrix[0])
        for i in range(rows):
            for j in range(i, cols):
                if matrix[i][j] != matrix[j][i]:
                    return False
        return True

    # Base Case: If the matrix is already symmetric, return 0
    if is_symmetric(matrix):
        return 0
    
    rows = len(matrix)
    cols = len(matrix[0])
    
    # Recursive Case 1: Check the leftmost and rightmost columns
    for i in range(rows):
        if matrix[i][0] != matrix[i][cols - 1]:
            # Insert a column
            new_col = [matrix[i][cols - 1] if matrix[i][0] == matrix[i][cols - 1] else min(matrix[i][0], matrix[i][cols - 1]) for i in range(len(matrix))]
            for i in range(len(matrix)):
                matrix[i].insert(0, new_col[i])
            return 1 + make_symmetric(matrix)
    
    # Recursive Case 2: Check the topmost and bottommost rows
    for j in range(cols):
        if matrix[0][j] != matrix[rows - 1][j]:
            # Insert a row
            new_row = [matrix[0][j] if matrix[0][j] == matrix[rows - 1][j] else min(matrix[0][j], matrix[rows - 1][j]) for j in range(len(matrix[0]))]
            matrix.insert(0, new_row)
            return 1 + make_symmetric(matrix)

# Example usage
matrix = [
    [1, 2, 3],
    [4, 5, 6],
    [3, 6, 7]
]
print(make_symmetric(matrix))

I've written a recursive Python function that attempts to solve the problem by making a matrix symmetric with a minimum number of row or column insertions. My function make_symmetric(matrix) has two recursive cases:

• It first checks the leftmost and rightmost columns to see if they are equal. If not, it inserts a column and recurses.
• Then it checks the topmost and bottommost rows to see if they are equal. If not, it inserts a row and recurses.

I expected that my function would return the minimum number of operations required to make the matrix symmetric. However, I've realized that my code doesn't contain any fail-safes for infinite loops, meaning it could keep inserting rows or columns endlessly without making the matrix symmetric.

Concerns:

• I know that I can improve the function by merging the is_symmetric() functionality directly into the if condition, but I'm not sure if that's the best approach.
• I'm considering modifying the code to make the matrix square first, but I'm unsure if that will minimize the number of insert operations.
• I'm not sure if inserting rows first, then columns, or alternating between them would result in fewer insertions to make the matrix symmetric.

I'm open to any suggestions or improvements for my code.