# Find trace of matrix formed by adding Row-major and Column-major order of same matrix

Given two integers N and M. Consider two matrix ANXM, BNXM. Both matrix A and matrix B contains elements from 1 to N*M. Matrix A contains elements in Row-major order and matrix B contains elements in Column-major order. The task is to find the trace of the matrix formed by addition of A and B. Trace of matrix PNXM is defined as P + P + P +….. + P[min(n – 1, m – 1)][min(n – 1, m – 1)] i.e addition of main diagonal.

Note – Both matrix A and matrix B contains elements from 1 to N*M.

Examples :

```Input : N = 3, M = 3
Output : 30
Therefore,
1 2 3
A = 4 5 6
7 8 9

1 4 7
B = 2 5 8
3 6 9

2 6 10
A + B = 6 10 14
10 14 18

Trace = 2 + 10 + 18 = 30
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Method 1 (Naive Approach) :
Generate matrix A and B and find the sum. Then traverse the main diagnol and find the sum.

Below is the implementation of this approach:

## C++

 `// C++ program to find ` `// trace of matrix formed by ` `// adding Row-major and ` `// Column-major order of same matrix ` `#include ` `using` `namespace` `std; ` ` `  `// Return the trace of ` `// sum of row-major matrix ` `// and column-major matrix ` `int` `trace(``int` `n, ``int` `m) ` `{ ` ` `  `    ``int` `A[n][m], B[n][m], C[n][m];     ` ` `  `    ``// Generating the matrix A ` `    ``int` `cnt = 1; ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``for` `(``int` `j = 0; j < m; j++) { ` `            ``A[i][j] = cnt; ` `            ``cnt++; ` `        ``}     ` ` `  `    ``// Generating the matrix A ` `    ``cnt = 1; ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``for` `(``int` `j = 0; j < m; j++) { ` `            ``B[j][i] = cnt; ` `            ``cnt++; ` `        ``} ` ` `  `    ``// Finding sum of matrix A and matrix B ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``for` `(``int` `j = 0; j < m; j++) ` `            ``C[i][j] = A[i][j] + B[i][j];     ` ` `  `    ``// Finding the trace of matrix C. ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``for` `(``int` `j = 0; j < m; j++) ` `            ``if` `(i == j) ` `                ``sum += C[i][j]; ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `N = 3, M = 3; ` `    ``cout << trace(N, M) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java program to find ` `// trace of matrix formed by ` `// adding Row-major and ` `// Column-major order of same matrix ` `class` `GFG ` `{ ` `    ``// Return the trace of ` `    ``// sum of row-major matrix ` `    ``// and column-major matrix ` `    ``static` `int` `trace(``int` `n, ``int` `m) ` `    ``{ ` `     `  `        ``int` `A[][] = ``new` `int``[n][m]; ` `        ``int` `B[][] = ``new` `int``[n][m]; ` `        ``int` `C[][] = ``new` `int``[n][m]; ` `     `  `        ``// Generating the matrix A ` `        ``int` `cnt = ``1``; ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``for` `(``int` `j = ``0``; j < m; j++) { ` `                ``A[i][j] = cnt; ` `                ``cnt++; ` `            ``}  ` `     `  `        ``// Generating the matrix A ` `        ``cnt = ``1``; ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``for` `(``int` `j = ``0``; j < m; j++) { ` `                ``B[j][i] = cnt; ` `                ``cnt++; ` `            ``} ` `     `  `        ``// Finding sum of matrix A and matrix B ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``for` `(``int` `j = ``0``; j < m; j++) ` `                ``C[i][j] = A[i][j] + B[i][j];  ` `     `  `        ``// Finding the trace of matrix C. ` `        ``int` `sum = ``0``; ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``for` `(``int` `j = ``0``; j < m; j++) ` `                ``if` `(i == j) ` `                    ``sum += C[i][j]; ` `     `  `        ``return` `sum; ` `    ``} ` `     `  `    ``// Driver code  ` `    ``public` `static` `void` `main (String[] args) ` `    ``{ ` `        ``int` `N = ``3``, M = ``3``; ` `         `  `        ``System.out.println(trace(N, M)); ` `    ``} ` `} ` ` `  `// This code is contributed by Anant Agarwal. `

## Python3

 `# Python3 program to find trace of matrix  ` `# formed by adding Row-major and ` `# Column-major order of same matrix ` ` `  `# Return the trace of sum of row-major  ` `# matrix and column-major matrix ` `def` `trace(n, m): ` ` `  `    ``A ``=` `[[``0` `for` `x ``in` `range``(m)]  ` `            ``for` `y ``in` `range``(n)]; ` `    ``B ``=` `[[``0` `for` `x ``in` `range``(m)]  ` `            ``for` `y ``in` `range``(n)]; ` `    ``C ``=` `[[``0` `for` `x ``in` `range``(m)]  ` `            ``for` `y ``in` `range``(n)];  ` ` `  `    ``# Generating the matrix A ` `    ``cnt ``=` `1``; ` `    ``for` `i ``in` `range``(n): ` `        ``for` `j ``in` `range``(m): ` `            ``A[i][j] ``=` `cnt; ` `            ``cnt ``+``=` `1``; ` ` `  `    ``# Generating the matrix A ` `    ``cnt ``=` `1``; ` `    ``for` `i ``in` `range``(n): ` `        ``for` `j ``in` `range``(m): ` `            ``B[j][i] ``=` `cnt; ` `            ``cnt ``+``=` `1``; ` ` `  `    ``# Finding sum of matrix A and matrix B ` `    ``for` `i ``in` `range``(n): ` `        ``for` `j ``in` `range``(m): ` `            ``C[i][j] ``=` `A[i][j] ``+` `B[i][j];  ` ` `  `    ``# Finding the trace of matrix C. ` `    ``sum` `=` `0``; ` `    ``for` `i ``in` `range``(n): ` `        ``for` `j ``in` `range``(m): ` `            ``if` `(i ``=``=` `j): ` `                ``sum` `+``=` `C[i][j]; ` ` `  `    ``return` `sum``; ` ` `  `# Driver Code ` `N ``=` `3``;  ` `M ``=` `3``; ` `print``(trace(N, M)); ` `     `  `# This code is contributed by mits `

## C#

 `// C# program to find ` `// trace of matrix formed by ` `// adding Row-major and ` `// Column-major order of same matrix ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Return the trace of ` `    ``// sum of row-major matrix ` `    ``// and column-major matrix ` `    ``static` `int` `trace(``int` `n, ``int` `m) ` `    ``{ ` `        ``int``[, ] A = ``new` `int``[n, m]; ` `        ``int``[, ] B = ``new` `int``[n, m]; ` `        ``int``[, ] C = ``new` `int``[n, m]; ` ` `  `        ``// Generating the matrix A ` `        ``int` `cnt = 1; ` `        ``for` `(``int` `i = 0; i < n; i++) ` `            ``for` `(``int` `j = 0; j < m; j++) { ` `                ``A[i, j] = cnt; ` `                ``cnt++; ` `            ``} ` ` `  `        ``// Generating the matrix A ` `        ``cnt = 1; ` `        ``for` `(``int` `i = 0; i < n; i++) ` `            ``for` `(``int` `j = 0; j < m; j++) { ` `                ``B[j, i] = cnt; ` `                ``cnt++; ` `            ``} ` ` `  `        ``// Finding sum of matrix A and matrix B ` `        ``for` `(``int` `i = 0; i < n; i++) ` `            ``for` `(``int` `j = 0; j < m; j++) ` `                ``C[i, j] = A[i, j] + B[i, j]; ` ` `  `        ``// Finding the trace of matrix C. ` `        ``int` `sum = 0; ` `        ``for` `(``int` `i = 0; i < n; i++) ` `            ``for` `(``int` `j = 0; j < m; j++) ` `                ``if` `(i == j) ` `                    ``sum += C[i, j]; ` ` `  `        ``return` `sum; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `N = 3, M = 3; ` `        ``Console.WriteLine(trace(N, M)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Output :

```30
```

Time Complexity: O(N*M).

Method 2 (efficient approach) :
Basically, we need to find the sum of main diagonal of the first matrix A and main diagonal of the second matrix B.
Let’s take an example, N = 3, M = 4.
Therefore, Row-major matrix will be,

```     1  2  3  4
A =  5  6  7  8
9 10 11 12
```

So, we need the sum of 1, 6, 11.
Observe, it form an Arithmetic Progression with constant difference of number of column, M.
Also, first element is always 1. So, AP formed in case of Row-major matrix is 1, 1+(M+1), 1+2*(M+1), ….. consisting of N (number of rows) elements. And we know,
Sn = (n * (a1 + an))/2
So, n = R, a1 = 1, an = 1 + (R – 1)*(M+1).

Similarly, in case of Column-major, AP formed will be 1, 1+(N+1), 1+2*(N+1), …..
So, n = R, a1 = 1, an = 1 + (R – 1)*(N+1).

Below is the implementation of this approach:

## C++

 `// C++ program to find trace of matrix formed ` `// by adding Row-major and Column-major order ` `// of same matrix ` `#include ` `using` `namespace` `std; ` ` `  `// Return sum of first n integers of an AP ` `int` `sn(``int` `n, ``int` `an) ` `{ ` `    ``return` `(n * (1 + an)) / 2; ` `} ` ` `  `// Return the trace of sum of row-major matrix ` `// and column-major matrix ` `int` `trace(``int` `n, ``int` `m) ` `{ ` `    ``// Finding nth element in ` `    ``// AP in case of Row major matrix. ` `    ``int` `an = 1 + (n - 1) * (m + 1); ` ` `  `    ``// Finding sum of first n integers ` `    ``// of AP in case of Row major matrix ` `    ``int` `rowmajorSum = sn(n, an); ` ` `  `    ``// Finding nth element in AP ` `    ``// in case of Row major matrix ` `    ``an = 1 + (n - 1) * (n + 1); ` ` `  `    ``// Finding sum of first n integers ` `    ``// of AP in case of Column major matrix ` `    ``int` `colmajorSum = sn(n, an); ` ` `  `    ``return` `rowmajorSum + colmajorSum; ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `N = 3, M = 3; ` `    ``cout << trace(N, M) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java program to find trace of matrix formed ` `// by adding Row-major and Column-major order ` `// of same matrix ` `import java.io.*; ` ` `  `public` `class` `GFG { ` ` `  `    ``// Return sum of first n integers of an AP ` `    ``static` `int` `sn(``int` `n, ``int` `an) ` `    ``{ ` `        ``return` `(n * (1 + an)) / 2; ` `    ``} ` ` `  `    ``// Return the trace of sum of row-major matrix ` `    ``// and column-major matrix ` `    ``static` `int` `trace(``int` `n, ``int` `m) ` `    ``{ ` `        ``// Finding nth element in ` `        ``// AP in case of Row major matrix. ` `        ``int` `an = 1 + (n - 1) * (m + 1); ` ` `  `        ``// Finding sum of first n integers ` `        ``// of AP in case of Row major matrix ` `        ``int` `rowmajorSum = sn(n, an); ` ` `  `        ``// Finding nth element in AP ` `        ``// in case of Row major matrix ` `        ``an = 1 + (n - 1) * (n + 1); ` ` `  `        ``// Finding sum of first n integers ` `        ``// of AP in case of Column major matrix ` `        ``int` `colmajorSum = sn(n, an); ` ` `  `        ``return` `rowmajorSum + colmajorSum; ` `    ``} ` ` `  `    ``// Driven Program ` `    ``static` `public` `void` `main(String[] args) ` `    ``{ ` `        ``int` `N = 3, M = 3; ` `        ``System.``out``.println(trace(N, M)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## Python3

 `# Python3 program to find trace  ` `# of matrix formed by adding ` `# Row-major and Column-major  ` `# order of same matrix ` ` `  `# Return sum of first n  ` `# integers of an AP ` `def` `sn(n, an): ` `    ``return` `(n ``*` `(``1` `+` `an)) ``/` `2``; ` ` `  `# Return the trace of sum ` `# of row-major matrix ` `# and column-major matrix ` `def` `trace(n, m): ` `     `  `    ``# Finding nth element  ` `    ``# in AP in case of ` `    ``# Row major matrix. ` `    ``an ``=` `1` `+` `(n ``-` `1``) ``*` `(m ``+` `1``); ` `     `  `    ``# Finding sum of first ` `    ``# n integers of AP in ` `    ``# case of Row major matrix ` `    ``rowmajorSum ``=` `sn(n, an); ` `     `  `    ``# Finding nth element in AP ` `    ``# in case of Row major matrix ` `    ``an ``=` `1` `+` `(n ``-` `1``) ``*` `(n ``+` `1``); ` `     `  `    ``# Finding sum of first n  ` `    ``# integers of AP in case  ` `    ``# of Column major matrix ` `    ``colmajorSum ``=` `sn(n, an); ` `     `  `    ``return` `int``(rowmajorSum ``+`  `               ``colmajorSum); ` `     `  `# Driver Code ` `N ``=` `3``; ` `M ``=` `3``; ` `print``(trace(N, M)); ` ` `  `# This code is contributed mits `

## C#

 `// C# program to find trace of matrix formed ` `// by adding Row-major and Column-major order ` `// of same matrix ` `using` `System; ` ` `  `public` `class` `GFG { ` ` `  `    ``// Return sum of first n integers of an AP ` `    ``static` `int` `sn(``int` `n, ``int` `an) ` `    ``{ ` `        ``return` `(n * (1 + an)) / 2; ` `    ``} ` ` `  `    ``// Return the trace of sum of row-major matrix ` `    ``// and column-major matrix ` `    ``static` `int` `trace(``int` `n, ``int` `m) ` `    ``{ ` `        ``// Finding nth element in ` `        ``// AP in case of Row major matrix. ` `        ``int` `an = 1 + (n - 1) * (m + 1); ` ` `  `        ``// Finding sum of first n integers ` `        ``// of AP in case of Row major matrix ` `        ``int` `rowmajorSum = sn(n, an); ` ` `  `        ``// Finding nth element in AP ` `        ``// in case of Row major matrix ` `        ``an = 1 + (n - 1) * (n + 1); ` ` `  `        ``// Finding sum of first n integers ` `        ``// of AP in case of Column major matrix ` `        ``int` `colmajorSum = sn(n, an); ` ` `  `        ``return` `rowmajorSum + colmajorSum; ` `    ``} ` ` `  `    ``// Driven Program ` `    ``static` `public` `void` `Main() ` `    ``{ ` `        ``int` `N = 3, M = 3; ` `        ``Console.WriteLine(trace(N, M)); ` `    ``} ` `} ` ` `  `// This code is contributed by vt_m. `

## PHP

 ` `

Output :

```30
```

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