Maximizing Matrix Sum: An Elegant Approach to Flip and Optimize
Intuition\nThe problem involves maximizing the sum of a matrix by flipping adjacent elements\' signs any number of times. The crucial insight is that flipping s
# Intuition\nThe problem involves maximizing the sum of a matrix by flipping adjacent elements\' signs any number of times. The crucial insight is that flipping signs does not affect the parity (odd/even nature) of the negative count. Hence, the strategy revolves around:\n\n1. Counting negatives to decide if a final adjustment is needed.\n2. Leveraging the smallest absolute value to handle odd negative counts.\n\n# Approach\n1. Initial Observations:\n    - Flipping signs of two adjacent numbers does not affect the total sum, as both are negated together.\n    - The matrix\'s sum is maximized when as many numbers as possible are positive.\n2. Key Steps:\n    - Calculate the total absolute sum of all matrix elements.\n    - Count the number of negative numbers.\n    - Find the smallest absolute value in the matrix, as it may need to be adjusted for an odd number of negatives.\n3. Result Derivation:\n    - If the number of negatives is even, all negatives can be paired up and flipped, so no adjustment is needed.\n    - If odd, one smallest absolute value must remain negative, so subtract twice its value to adjust.\n\n# Complexity\n- Time complexity:\n$$O(m*n)$$\n\n- Space complexity:\n$$O(1)$$\n\n# Code\n```python3 []\nclass Solution:\n    def maxMatrixSum(self, matrix: List[List[int]]) -> int:\n        negCount = total = 0\n        minValue = inf # To store the smallest absolute value\n        for row in matrix:\n            for col in row:\n                val = col\n                if col < 0:\n                    negCount += 1 # Count negative numbers\n                    val *= -1\n                if val < minValue:\n                    minValue = val\n                total += val\n        # Adjust total if odd negatives\n        return total - (minValue * 2) if negCount % 2 else total\n\n```
['Array', 'Greedy', 'Matrix', 'Python3']