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Maximum Matrix Sum

Difficulty: Medium

Problem Description

You are given an n x n integer matrix. You can do the following operation any number of times: Choose any two adjacent elements of the matrix and multiply each of them by -1. Your goal is to maximize the summation of the matrix's elements. Return the maximum sum of the matrix's elements using the operation mentioned above.

Key Insights

  • The operation allows you to change the signs of two adjacent elements, effectively allowing you to manipulate the matrix to maximize the sum.
  • The total sum of the matrix can be maximized by ensuring that as many negative values as possible are converted to positive values.
  • The absolute value of the sum of all elements will be maximized, considering the ability to flip the signs of adjacent elements.

Space and Time Complexity

Time Complexity: O(n^2)
Space Complexity: O(1)

Solution

The strategy involves calculating the total absolute sum of all the elements in the matrix. By flipping the signs of negative elements, we can maximize the overall matrix sum. The process includes:

  1. Iterating through the matrix to calculate the total sum of absolute values.
  2. Checking how many negative elements exist to determine if we need to adjust the final sum.
  3. If there is an odd number of negative elements, we will need to subtract the smallest absolute value from the total sum to account for one element that cannot be flipped.

Code Solutions

def maxMatrixSum(matrix):
    total_sum = 0
    min_abs = float('inf')
    negative_count = 0

    for row in matrix:
        for value in row:
            total_sum += abs(value)
            if value < 0:
                negative_count += 1
            min_abs = min(min_abs, abs(value))

    # If the count of negative numbers is odd, reduce the total sum by twice the smallest absolute value
    if negative_count % 2 == 1:
        total_sum -= 2 * min_abs

    return total_sum
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