**π‘ Problem Formulation:** The challenge is to determine if, for a given i-th row and i-th column in a square matrix, their respective sums are equal. Specifically, we are interested in knowing whether the sum of the elements in a row is the same as the sum of the elements in the corresponding column for each index i. For instance, given the matrix `[[1, 2], [3, 4]]`

, we want to verify this condition for rows and columns at index 0 and 1 (`1+2 ? 1+3`

and `3+4 ? 2+4`

respectively).

## Method 1: Iterative Comparison

This method involves iterating through each index of the matrix rows and columns to calculate their sums individually and then checking if they are equal. We would use two nested loops, the outer loop for the rows and the inner one for the columns, and then we would compare the sums at each iteration.

Here’s an example:

def check_sums(matrix): for i in range(len(matrix)): if sum(matrix[i]) != sum(row[i] for row in matrix): return False return True matrix_example = [[1, 2], [3, 4]] print(check_sums(matrix_example))

Output: `False`

This snippet defines a function `check_sums()`

that takes a matrix as input and checks if for all indices ‘i’, the sum of elements in the i-th row is equal to the sum of elements in the i-th column. It compares them by calculating the row sum with `sum(matrix[i])`

and column sum with a generator expression inside the sum function. The function returns `False`

as soon as the equivalence fails for any index, otherwise `True`

after all comparisons.

## Method 2: Using List Comprehension and zip

The zip function can be used in combination with list comprehension to compare row and column sums in a more Pythonic way. The zip function pairs elements from multiple iterators (like rows of a matrix), which can then be summed iteratively.

Here’s an example:

def check_sums(matrix): return all(sum(row) == sum(col) for row, col in zip(matrix, zip(*matrix))) matrix_example = [[1, 2], [3, 4]] print(check_sums(matrix_example))

Output: `False`

In this code snippet, `check_sums()`

uses a single list comprehension to handle both the rows and columns sum. The clever use of `zip(*matrix)`

transposes the matrix to compare column sums directly against row sums. The result is a neater one-liner that returns a boolean value based on whether all row and column sums are equivalent.

## Method 3: NumPy Library

Using the NumPy library, which is optimized for numerical operations on arrays, we can achieve our goal with less code and potentially better performance. This method assumes the matrix is a NumPy array for sum operations along different axes.

Here’s an example:

import numpy as np def check_sums(matrix): return np.all(np.sum(matrix, axis=1) == np.sum(matrix, axis=0)) matrix_example = np.array([[1, 2], [3, 4]]) print(check_sums(matrix_example))

Output: `False`

In this case, `check_sums()`

utilizes NumPy’s `sum()`

function, specifying the axis of summation (0 for columns and 1 for rows). The equality of the arrays containing the sums is checked with `np.all()`

, returning a single boolean value that indicates if all sums are equal.

## Method 4: Map and Lambda Functions

Using map and lambda functions, we apply a more functional programming approach to the problem. The map function can transform each row into its sum, while the lambda function computes the column sums.

Here’s an example:

def check_sums(matrix): col_sums = map(lambda *row: sum(row), *matrix) row_sums = map(sum, matrix) return all(r == c for r, c in zip(row_sums, col_sums)) matrix_example = [[1, 2], [3, 4]] print(check_sums(matrix_example))

Output: `False`

The `check_sums()`

function takes advantage of `map()`

for applying the sum function to each row and a lambda function for each column. The results are then paired using zip and checked for equality within a single iteration. The all() function again makes it concise by consolidating the results into a single boolean output.

## Bonus One-Liner Method 5: Using Set and Generator Expression

For a compact and efficient one-liner solution, we can use a set to compare the sums of each row and column in a matrix, leveraging generator expressions.

Here’s an example:

check_sums = lambda m: len({sum(r) for r in m}.union({sum(m[i][j] for i in range(len(m))) for j in range(len(m[0]))})) == 1 matrix_example = [[1, 2], [3, 4]] print(check_sums(matrix_example))

Output: `False`

This one-liner defines a lambda function that creates two sets from generator expressions, one for row sums and another for column sums, and then checks if their union results in a set of only one element, indicating that all sums are identical. It’s a succinct solution but perhaps less readable for those unfamiliar with Python’s advanced set and generator features.

## Summary/Discussion

**Method 1:**Iterative Comparison. Simple and straightforward. It might be less efficient with large matrices.

**Method 2:**Using List Comprehension and zip. Elegant and Pythonic. It could be confusing for beginners due to the transformation of the matrix.

**Method 3:**NumPy Library. Best for performance on large matrices. Requires an external library (NumPy), which is not always feasible.

**Method 4:**Map and Lambda Functions. Showcases functional programming in Python. It could be less intuitive for those not familiar with functional programming concepts.

**Bonus Method 5:**Using Set and Generator Expression. Extremely succinct and performs well. The compactness might sacrifice readability and maintainability.