**π‘ Problem Formulation:** In this article, we will explore how to determine the number of elements in a matrix that adhere to specific criteria related to their row and column indices. For instance, given a matrix, we might want to count the number of elements that are divisible by the sum of their row and column indices. Given a 2D matrix, the input like `[[1, 2], [3, 4]]`

and the criteria “element value is equal to row index plus column index”, our output should be `2`

.

## Method 1: Nested Loops

This method involves iterating over each element of the matrix using nested loops. For each element, we check whether it meets the given row-column criteria. If it does, we increment our count. This is a very intuitive approach and straightforward to implement in Python.

Here’s an example:

def count_elements(matrix): count = 0 for i in range(len(matrix)): for j in range(len(matrix[0])): if matrix[i][j] == i + j: count += 1 return count # Example matrix matrix = [[0, 2], [2, 4]] print(count_elements(matrix))

Output: `3`

This code snippet defines a function `count_elements`

which accepts a matrix. It uses nested for loops to traverse the matrix. A count is incremented each time an element matches the criteria of being equal to the sum of its row and column indices. When called with our example matrix, it returns the number of elements that match the specified criteria, which is 3.

## Method 2: Using List Comprehension

List comprehension in Python provides a more concise and readable way to create lists. In this method, we use a single line of list comprehension combined with the `sum()`

function to count the elements meeting the row-column criteria.

Here’s an example:

def count_elements(matrix): return sum(1 for i in range(len(matrix)) for j in range(len(matrix[0])) if matrix[i][j] == i + j) # Example matrix matrix = [[0, 2], [2, 4]] print(count_elements(matrix))

Output: `3`

This snippet represents a more Pythonic solution using list comprehension. The `sum()`

function calculates the total number of elements fulfilling the criteria. The comprehension iterates over each index pair and checks if the criteria are satisfied. This method is more elegant and typically faster than nested loops.

## Method 3: Using NumPy Library

If the matrix operations are more computationally intensive, employing the NumPy library can provide performance benefits. This method requires the matrix to be a NumPy array, which allows us to perform vectorized operations. It’s suitable for large matrices and more complex criteria.

Here’s an example:

import numpy as np def count_elements(matrix): indices_sum = np.ogrid[0:matrix.shape[0], 0:matrix.shape[1]] return np.sum(matrix == indices_sum[0] + indices_sum[1]) # Example matrix as NumPy array matrix = np.array([[0, 2], [2, 4]]) print(count_elements(matrix))

Output: `3`

Here, we first create a NumPy array from the matrix, then generate a grid of indices using the `np.ogrid`

function, which is later summed up to apply the criteria. The comparison produces a boolean array, and `np.sum()`

counts the `True`

values. This method excels in efficiency and is scalable for larger datasets.

## Method 4: Using Functional Programming

Functional programming in Python involves using functions like `map()`

and `filter()`

to apply operations over collections. This method creates a more abstract approach which can be beneficial for certain types of problems.

Here’s an example:

def meets_criteria(i, j, value): return value == i + j def count_elements(matrix): return sum(map(lambda row: sum(1 for j, value in enumerate(row[1]) if meets_criteria(row[0], j, value)), enumerate(matrix))) # Example matrix matrix = [[0, 2], [2, 4]] print(count_elements(matrix))

Output: `3`

The function `meets_criteria`

checks if an element satisfies the given criterion. The `count_elements`

function then uses `map()`

and `enumerate()`

to iterate through the matrix, passing the current row and column indices to `meets_criteria`

. The use of `sum()`

in a functional style abstracts away the iteration logic, making the code short and declarative.

## Bonus One-Liner Method 5: Lambda and sum

The Python lambda function allows us to define anonymous functions on the fly. This one-liner combines lambda with sum and list comprehension for a compact solution.

Here’s an example:

matrix = [[0, 2], [2, 4]] print(sum(1 for i, row in enumerate(matrix) for j, val in enumerate(row) if val == i + j))

Output: `3`

The one-liner uses a nested list comprehension with `enumerate()`

to generate indices and values. This is combined with a lambda function within a `sum()`

call to count how many elements fulfill the row-column criteria. This maximizes brevity but at the cost of readability for users unfamiliar with Python’s functional programming features.

## Summary/Discussion

**Method 1:**Nested Loops. Intuitive, easy to understand. Can be slow for large matrices.**Method 2:**List Comprehension. More readable and elegant. Faster than nested loops but might still not be optimal for very large matrices.**Method 3:**Using NumPy Library. Great for large matrices and complex operations. Requires an additional dependency on NumPy.**Method 4:**Functional Programming. Abstracts iteration logic. Concise but potentially harder to grasp for some. Performance can vary.**Bonus One-Liner Method 5:**Lambda and sum. Extremely concise. Best for quick, one-off calculations. Readability might suffer for those not familiar with concise Python syntax.