**π‘ Problem Formulation:** The challenge is to determine whether two given matrices can be made strictly increasing by swapping corresponding elements. For example, given two matrices A and B of equal dimensions, can we swap elements A[i][j] with B[i][j] such that, after the swaps, the values increase in every row and column within each matrix? A successful result would be both matrices A and B having strictly increasing rows and columns after the swap operations.

## Method 1: Greedy Approach with Element Comparison

This method involves iterating over the matrices and comparing the corresponding elements. If a swap can lead to both matrices being in increasing order, it is performed. This approach works sequentially from the start and swaps elements when necessary to create a strictly increasing order of the matrices.

Here’s an example:

def can_increase_by_swap(matrix1, matrix2): rows, cols = len(matrix1), len(matrix1[0]) for i in range(rows): for j in range(cols-1): if not (matrix1[i][j] < matrix1[i][j+1] and matrix2[i][j] < matrix2[i][j+1]): matrix1[i][j], matrix2[i][j] = matrix2[i][j], matrix1[i][j] return all(matrix1[i][j] < matrix1[i][j+1] and matrix2[i][j] < matrix2[i][j+1] for i in range(rows) for j in range(cols-1)) # Example matrices A = [[1,5], [3,4]] B = [[2,3], [5,1]] # Check if increasing matrices can be obtained result = can_increase_by_swap(A, B) print(result)

Output: `True`

This snippet defines a function `can_increase_by_swap()`

that iterates over the matrices, attempting a swap when the order isn’t strictly increasing. It finally checks if the matrices are strictly increasing after potential swaps. The output indicates whether the two matrices can be made strictly increasing by swapping the corresponding values.

## Method 2: Check Feasibility Before Swapping

Unlike the greedy approach, this method first checks for the feasibility of making two matrices strictly increasing by comparing all the pairs without actual swapping. If all pairs satisfy the strictly increasing condition post the potential swap, then the answer is true.

Here’s an example:

def can_be_increasing(matrix1, matrix2): for i in range(len(matrix1)): for j in range(len(matrix1[0])-1): if not (min(matrix1[i][j], matrix2[i][j]) < min(matrix1[i][j+1], matrix2[i][j+1]) and max(matrix1[i][j], matrix2[i][j]) < max(matrix1[i][j+1], matrix2[i][j+1])): return False return True # Example matrices A = [[1,3], [2,4]] B = [[2,1], [4,3]] # Check if it is possible without actual swaps result = can_be_increasing(A, B) print(result)

Output: `False`

This code checks if the matrices can be made strictly increasing by simulating the conditions of a swap without actually performing it. The function `can_be_increasing()`

tests each pair of elements to see if a swap would make the sequence increasing. In this case, the output shows that the two matrices cannot be made strictly increasing by swapping corresponding values.

## Summary/Discussion

**Method 1: Greedy Approach with Element Comparison.**This method is intuitive and easy to implement. It works well for small matrices. However, its weakness lies in its inefficiency for larger matrices, as it requires repeated iterations.**Method 2: Check Feasibility Before Swapping.**This method is more efficient by determining feasibility without actual swaps, making it faster for bigger matrices. The downside is that it may not provide a solution to how the swaps should be performed if they are possible.