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# Minimum operations required to make each row and column of matrix equals

• Difficulty Level : Medium
• Last Updated : 05 May, 2021

Given a square matrix of size . Find minimum number of operation are required such that sum of elements on each row and column becomes equals. In one operation, increment any value of cell of matrix by 1. In first line print minimum operation required and in next ‘n’ lines print ‘n’ integers representing the final matrix after operation.
Example:

Input:
1 2
3 4
Output:
4
4 3
3 4
Explanation
1. Increment value of cell(0, 0) by 3
2. Increment value of cell(0, 1) by 1
Hence total 4 operation are required

Input: 9
1 2 3
4 2 3
3 2 1
Output:
6
2 4 3
4 2 3
3 3 3 

The approach is simple, let’s assume that maxSum is the maximum sum among all rows and columns. We just need to increment some cells such that the sum of any row or column becomes ‘maxSum’.
Let’s say Xi is the total number of operation needed to make the sum on row ‘i’ equals to maxSum and Yj is the total number of operation needed to make the sum on column ‘j’ equals to maxSum. Since Xi = Yj so we need to work at any one of them according to the condition.
In order to minimise Xi, we need to choose the maximum from rowSumi and colSumj as it will surely lead to minimum operation. After that, increment ‘i’ or ‘j’ according to the condition satisfied after increment.
Below is the implementation of the above approach.

## C++

 /* C++ Program to Find minimum number ofoperation required such that sum ofelements on each row and column becomes same*/#include using namespace std; // Function to find minimum operation required// to make sum of each row and column equalsint findMinOpeartion(int matrix[][2], int n){    // Initialize the sumRow[] and sumCol[]    // array to 0    int sumRow[n], sumCol[n];    memset(sumRow, 0, sizeof(sumRow));    memset(sumCol, 0, sizeof(sumCol));     // Calculate sumRow[] and sumCol[] array    for (int i = 0; i < n; ++i)        for (int j = 0; j < n; ++j) {            sumRow[i] += matrix[i][j];            sumCol[j] += matrix[i][j];        }     // Find maximum sum value in either    // row or in column    int maxSum = 0;    for (int i = 0; i < n; ++i) {        maxSum = max(maxSum, sumRow[i]);        maxSum = max(maxSum, sumCol[i]);    }     int count = 0;    for (int i = 0, j = 0; i < n && j < n;) {         // Find minimum increment required in        // either row or column        int diff = min(maxSum - sumRow[i],                       maxSum - sumCol[j]);         // Add difference in corresponding cell,        // sumRow[] and sumCol[] array        matrix[i][j] += diff;        sumRow[i] += diff;        sumCol[j] += diff;         // Update the count variable        count += diff;         // If ith row satisfied, increment ith        // value for next iteration        if (sumRow[i] == maxSum)            ++i;         // If jth column satisfied, increment        // jth value for next iteration        if (sumCol[j] == maxSum)            ++j;    }    return count;} // Utility function to print matrixvoid printMatrix(int matrix[][2], int n){    for (int i = 0; i < n; ++i) {        for (int j = 0; j < n; ++j)            cout << matrix[i][j] << " ";        cout << "\n";    }} // Driver codeint main(){    int matrix[][2] = { { 1, 2 },                        { 3, 4 } };    cout << findMinOpeartion(matrix, 2) << "\n";    printMatrix(matrix, 2);    return 0;}

## Java

 // Java Program to Find minimum// number of operation required// such that sum of elements on// each row and column becomes sameimport java.io.*; class GFG {     // Function to find minimum    // operation required    // to make sum of each row    // and column equals    static int findMinOpeartion(int matrix[][],                                         int n)    {        // Initialize the sumRow[]        // and sumCol[] array to 0        int[] sumRow = new int[n];        int[] sumCol = new int[n];                 // Calculate sumRow[] and        // sumCol[] array        for (int i = 0; i < n; ++i)              for (int j = 0; j < n; ++j)            {                sumRow[i] += matrix[i][j];                sumCol[j] += matrix[i][j];            }             // Find maximum sum value        // in either row or in column        int maxSum = 0;        for (int i = 0; i < n; ++i)        {            maxSum = Math.max(maxSum, sumRow[i]);            maxSum = Math.max(maxSum, sumCol[i]);        }             int count = 0;        for (int i = 0, j = 0; i < n && j < n;)        {            // Find minimum increment            // required in either row            // or column            int diff = Math.min(maxSum - sumRow[i],                        maxSum - sumCol[j]);                 // Add difference in            // corresponding cell,            // sumRow[] and sumCol[]            // array            matrix[i][j] += diff;            sumRow[i] += diff;            sumCol[j] += diff;                 // Update the count            // variable            count += diff;                 // If ith row satisfied,            // increment ith value            // for next iteration            if (sumRow[i] == maxSum)                ++i;                 // If jth column satisfied,            // increment jth value for            // next iteration            if (sumCol[j] == maxSum)                ++j;        }        return count;    }     // Utility function to    // print matrix    static void printMatrix(int matrix[][],                                     int n)    {        for (int i = 0; i < n; ++i)        {            for (int j = 0; j < n; ++j)                System.out.print(matrix[i][j] +                                           " ");                      System.out.println();        }    }     /* Driver program */    public static void main(String[] args)    {        int matrix[][] = {{1, 2},                          {3, 4}};                 System.out.println(findMinOpeartion(matrix, 2));        printMatrix(matrix, 2);     }} // This code is contributed by Gitanjali.

## Python 3

 # Python 3 Program to Find minimum# number of operation required such# that sum of elements on each row# and column becomes same # Function to find minimum operation# required to make sum of each row# and column equalsdef findMinOpeartion(matrix, n):     # Initialize the sumRow[] and sumCol[]    # array to 0    sumRow = [0] * n    sumCol = [0] * n     # Calculate sumRow[] and sumCol[] array    for i in range(n):        for j in range(n) :            sumRow[i] += matrix[i][j]            sumCol[j] += matrix[i][j]     # Find maximum sum value in    # either row or in column    maxSum = 0    for i in range(n) :        maxSum = max(maxSum, sumRow[i])        maxSum = max(maxSum, sumCol[i])     count = 0    i = 0    j = 0    while i < n and j < n :         # Find minimum increment required        # in either row or column        diff = min(maxSum - sumRow[i],                   maxSum - sumCol[j])         # Add difference in corresponding        # cell, sumRow[] and sumCol[] array        matrix[i][j] += diff        sumRow[i] += diff        sumCol[j] += diff         # Update the count variable        count += diff         # If ith row satisfied, increment        # ith value for next iteration        if (sumRow[i] == maxSum):            i += 1         # If jth column satisfied, increment        # jth value for next iteration        if (sumCol[j] == maxSum):            j += 1                 return count # Utility function to print matrixdef printMatrix(matrix, n):    for i in range(n) :        for j in range(n):            print(matrix[i][j], end = " ")        print() # Driver codeif __name__ == "__main__":    matrix = [[ 1, 2 ],              [ 3, 4 ]]    print(findMinOpeartion(matrix, 2))    printMatrix(matrix, 2) # This code is contributed# by ChitraNayal

## C#

 // C# Program to Find minimum// number of operation required// such that sum of elements on// each row and column becomes sameusing System; class GFG {     // Function to find minimum    // operation required    // to make sum of each row    // and column equals    static int findMinOpeartion(int [,]matrix,                                        int n)    {        // Initialize the sumRow[]        // and sumCol[] array to 0        int[] sumRow = new int[n];        int[] sumCol = new int[n];                 // Calculate sumRow[] and        // sumCol[] array        for (int i = 0; i < n; ++i)             for (int j = 0; j < n; ++j)            {                sumRow[i] += matrix[i,j];                sumCol[j] += matrix[i,j];            }             // Find maximum sum value        // in either row or in column        int maxSum = 0;        for (int i = 0; i < n; ++i)        {            maxSum = Math.Max(maxSum, sumRow[i]);            maxSum = Math.Max(maxSum, sumCol[i]);        }             int count = 0;        for (int i = 0, j = 0; i < n && j < n;)        {            // Find minimum increment            // required in either row            // or column            int diff = Math.Min(maxSum - sumRow[i],                        maxSum - sumCol[j]);                 // Add difference in            // corresponding cell,            // sumRow[] and sumCol[]            // array            matrix[i,j] += diff;            sumRow[i] += diff;            sumCol[j] += diff;                 // Update the count            // variable            count += diff;                 // If ith row satisfied,            // increment ith value            // for next iteration            if (sumRow[i] == maxSum)                ++i;                 // If jth column satisfied,            // increment jth value for            // next iteration            if (sumCol[j] == maxSum)                ++j;        }        return count;    }     // Utility function to    // print matrix    static void printMatrix(int [,]matrix,                                    int n)    {        for (int i = 0; i < n; ++i)        {            for (int j = 0; j < n; ++j)                Console.Write(matrix[i,j] +                                        " ");                     Console.WriteLine();        }    }     /* Driver program */    public static void Main()    {        int [,]matrix = {{1, 2},                        {3, 4}};                 Console.WriteLine(findMinOpeartion(matrix, 2));        printMatrix(matrix, 2);     }} // This code is contributed by Vt_m.

## Javascript

 
Output
4
4 3
3 4

Time complexity: O(n2
Auxiliary space: O(n)

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