# Maximize the sum of Kth column of a Matrix

Given two integers N and K, the task is to maximize the sum of the Kth column of N * N row-wise sorted matrix consisting of element in the range [1, N2].

Examples:

Input: N = 2, K = 2
Output: {{1, 3}, {2, 4}}
Explanations: The possible row-wise sorted matrices are [{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}, {{3, 4}, {1, 2}}, {{2, 4}, {1, 3}}, {{2, 3}, {1, 4}} ]
Out of all the above possible matrices, the matrices [{{1, 3}, {2, 4}}, {{2, 4}, {1, 3}}, {{1, 4}, {2, 3}}, {{2, 3}, {1, 4}}] contains the maximum possible sum of the Kth column.
Therefore, one of the possible output is {{1, 3}, {2, 4}}.

Input: N = 3, K = 2
Output: {{1, 4, 5}, {2, 6, 7}, {3, 8, 9}}

Approach: The idea here is to first fill the indices smaller than the Kth columns of the matrix by the values from the range [1, N * (K – 1)] and then fill all the elements at columns greater than or equal to the kth column by values from the range [N * (K – 1) + 1, N2] as shown in the image below. Follow the steps below to solve the problem:

1. Fill all the columns of the matrix smaller than K by the values from the range [1, N * (K – 1)].
2. Then, fill the columns of the matrix greater than or equal to K by the values from the range [N * (K – 1) + 1, N * N].
3. Finally, print the matrix.

Below is the implementation of the above approach:

## C++

 `// C++ program to implement ` `// the above approach ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to maximize the Kth column sum ` `int``** findMatrix(``int` `N, ``int` `K) ` `{ ` ` `  `    ``// Store all the elements of the ` `    ``// resultant matrix of size N*N ` `    ``int``** mat = (``int``**)``malloc``( ` `        ``N * ``sizeof``(``int``*)); ` ` `  `    ``for` `(``int` `i = 0; i < N; ++i) { ` `        ``mat[i] = (``int``*)``malloc``( ` `            ``N * ``sizeof``(``int``)); ` `    ``} ` ` `  `    ``// Store value of each ` `    ``// elements of the matrix ` `    ``int` `element = 1; ` ` `  `    ``// Fill all the columns < K ` `    ``for` `(``int` `i = 0; i < N; ++i) { ` ` `  `        ``for` `(``int` `j = 0; j < K - 1; ++j) { ` `            ``mat[i][j] = element++; ` `        ``} ` `    ``} ` ` `  `    ``// Fill all the columns >= K ` `    ``for` `(``int` `i = 0; i < N; ++i) { ` ` `  `        ``for` `(``int` `j = K - 1; j < N; ++j) { ` `            ``mat[i][j] = element++; ` `        ``} ` `    ``} ` ` `  `    ``return` `mat; ` `} ` ` `  `// Function to print the matrix ` `void` `printMatrix(``int``** mat, ``int` `N) ` `{ ` ` `  `    ``for` `(``int` `i = 0; i < N; ++i) { ` `        ``for` `(``int` `j = 0; j < N; ++j) { ` `            ``cout << mat[i][j] << ``" "``; ` `        ``} ` `        ``cout << endl; ` `    ``} ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` ` `  `    ``int` `N = 3, K = 2; ` `    ``int``** mat = findMatrix(N, K); ` ` `  `    ``printMatrix(mat, N); ` `}`

## Java

 `// Java program to implement ` `// the above approach ` `class` `GFG{ ` ` `  `// Function to maximize the Kth column sum ` `static` `int` `[][]findMatrix(``int` `N, ``int` `K) ` `{ ` ` `  `    ``// Store all the elements of the ` `    ``// resultant matrix of size N*N ` `    ``int` `[][]mat = ``new` `int``[N][N]; ` ` `  `    ``// Store value of each ` `    ``// elements of the matrix ` `    ``int` `element = ``1``; ` ` `  `    ``// Fill all the columns < K ` `    ``for``(``int` `i = ``0``; i < N; ++i) ` `    ``{ ` `        ``for``(``int` `j = ``0``; j < K - ``1``; ++j) ` `        ``{ ` `            ``mat[i][j] = element++; ` `        ``} ` `    ``} ` ` `  `    ``// Fill all the columns >= K ` `    ``for``(``int` `i = ``0``; i < N; ++i)  ` `    ``{ ` `        ``for``(``int` `j = K - ``1``; j < N; ++j) ` `        ``{ ` `            ``mat[i][j] = element++; ` `        ``} ` `    ``} ` `    ``return` `mat; ` `} ` ` `  `// Function to print the matrix ` `static` `void` `printMatrix(``int` `[][]mat, ``int` `N) ` `{ ` `    ``for``(``int` `i = ``0``; i < N; ++i)  ` `    ``{ ` `        ``for``(``int` `j = ``0``; j < N; ++j) ` `        ``{ ` `            ``System.out.print(mat[i][j] + ``" "``); ` `        ``} ` `        ``System.out.println(); ` `    ``} ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `N = ``3``, K = ``2``; ` `    ``int` `[][]mat = findMatrix(N, K); ` ` `  `    ``printMatrix(mat, N); ` `} ` `} ` ` `  `// This code is contributed by Amit Katiyar`

## Python3

 `# Python3 program to implement ` `# the above approach ` ` `  `# Function to maximize the Kth  ` `# column sum ` `def` `findMatrix(N, K): ` `     `  `    ``# Store all the elements of the ` `    ``# resultant matrix of size N*N ` `    ``mat ``=` `[[``0` `for` `i ``in` `range``(N)] ` `              ``for` `j ``in` `range``(N)]; ` ` `  `    ``# Store value of each ` `    ``# elements of the matrix ` `    ``element ``=` `0``; ` ` `  `    ``# Fill all the columns < K ` `    ``for` `i ``in` `range``(``0``, N): ` `        ``for` `j ``in` `range``(``0``, K ``-` `1``): ` `            ``element ``+``=` `1``; ` `            ``mat[i][j] ``=` `element; ` ` `  `    ``# Fill all the columns >= K ` `    ``for` `i ``in` `range``(``0``, N): ` `        ``for` `j ``in` `range``(K ``-` `1``, N): ` `            ``element ``+``=` `1``; ` `            ``mat[i][j] ``=` `element; ` ` `  `    ``return` `mat; ` ` `  `# Function to prthe matrix ` `def` `printMatrix(mat, N): ` `     `  `    ``for` `i ``in` `range``(``0``, N): ` `        ``for` `j ``in` `range``(``0``, N): ` `            ``print``(mat[i][j], end ``=` `" "``); ` ` `  `        ``print``(); ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `'__main__'``: ` `     `  `    ``N ``=` `3``; K ``=` `2``; ` `    ``mat ``=` `findMatrix(N, K); ` ` `  `    ``printMatrix(mat, N); ` ` `  `# This code is contributed by Amit Katiyar`

## C#

 `// C# program to implement ` `// the above approach ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `// Function to maximize the Kth column sum ` `static` `int` `[,]findMatrix(``int` `N, ``int` `K) ` `{ ` ` `  `    ``// Store all the elements of the ` `    ``// resultant matrix of size N*N ` `    ``int` `[,]mat = ``new` `int``[N, N]; ` ` `  `    ``// Store value of each ` `    ``// elements of the matrix ` `    ``int` `element = 1; ` ` `  `    ``// Fill all the columns < K ` `    ``for``(``int` `i = 0; i < N; ++i) ` `    ``{ ` `        ``for``(``int` `j = 0; j < K - 1; ++j) ` `        ``{ ` `            ``mat[i, j] = element++; ` `        ``} ` `    ``} ` ` `  `    ``// Fill all the columns >= K ` `    ``for``(``int` `i = 0; i < N; ++i)  ` `    ``{ ` `        ``for``(``int` `j = K - 1; j < N; ++j) ` `        ``{ ` `            ``mat[i, j] = element++; ` `        ``} ` `    ``} ` `    ``return` `mat; ` `} ` ` `  `// Function to print the matrix ` `static` `void` `printMatrix(``int` `[,]mat, ``int` `N) ` `{ ` `    ``for``(``int` `i = 0; i < N; ++i)  ` `    ``{ ` `        ``for``(``int` `j = 0; j < N; ++j) ` `        ``{ ` `            ``Console.Write(mat[i, j] + ``" "``); ` `        ``} ` `        ``Console.WriteLine(); ` `    ``} ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `N = 3, K = 2; ` `    ``int` `[,]mat = findMatrix(N, K); ` ` `  `    ``printMatrix(mat, N); ` `} ` `} ` ` `  `// This code is contributed by Amit Katiyar`

Output:

```1 4 5
2 6 7
3 8 9
```

Time Complexity: O(N2)
Auxiliary Space: O(N2)

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