# Find determinant of matrix generated by array rotation

Given an array of three elements. The task is to construct a matrix of order 3Ã—3 by using all three rotations of the array as a row of the matrix and find the determinant of the resultant matrix.

Examples:

```Input : arr[] = {1, 2, 3}
Output : 18

Input : arr[] = {1, 1, 1}
Output : 0```

Approach: As per the problem statement, construct a 3*3 matrix using the given array. If a1, a2, a3 are array elements then the corresponding matrix will be:

```{{a1, a2, a3},
{a3, a1, a2},
{a2, a3, a1}}```

The task is to calculate the determinant of the above matrix.
The determinant can be calculated by using the proper method for this but on the other hand, if the resultant matrix is expanded for calculation, the result will be a13 + a23 + a33 – 3*a1*a2*a3. Hence, instead of calculating determinants by proper expansion use the above-generated formula.
Therefore, the Determinant of the above Matrix will be:

`a13 + a23 + a33 - (3*a1*a2*a3)`

Below is the implementation of the above approach:

## C++

 `// C++ program for finding determinant of generated matrix`   `#include ` `#define N 3` `using` `namespace` `std;`   `// Function to calculate determinant` `int` `calcDeterminant(``int` `arr[])` `{` `    ``int` `determinant = 0;`   `    ``for` `(``int` `i = 0; i < N; i++) {` `        ``determinant += ``pow``(arr[i], 3);` `    ``}`   `    ``determinant -= 3 * arr[0] * arr[1] * arr[2];`   `    ``return` `determinant;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `arr[] = { 4, 5, 3 };` `    ``cout << calcDeterminant(arr);` `    ``return` `0;` `}`

## Java

 `// Java program for finding determinant` `// of generated matrix ` `import` `java.util.*;` `import` `java.lang.*;`   `class` `GFG` `{` `static` `int` `N = ``3``;`   `// Function to calculate determinant ` `static` `double` `calcDeterminant(``int` `arr[]) ` `{ ` `    ``double` `determinant = ``0``; `   `    ``for` `(``int` `i = ``0``; i < N; i++) ` `    ``{ ` `        ``determinant += Math.pow(arr[i], ``3``); ` `    ``} `   `    ``determinant -= ``3` `* arr[``0``] * ` `                    ``arr[``1``] * arr[``2``]; `   `    ``return` `determinant; ` `} `   `// Driver code` `static` `public` `void` `main (String args[])` `{` `    ``int` `[]arr = { ``4``, ``5``, ``3` `}; ` `    ``System.out.println(calcDeterminant(arr)); ` `}` `}`   `// This code is contributed ` `// by Akanksha Rai`

## Python3

 `# Python3 program for finding determinant of generated matrix`   `# Function to calculate determinant` `def` `calcDeterminant(arr,n):` `    ``determinant ``=``0`   `    ``for` `i ``in` `range``(n):` `        ``determinant``+``=` `pow``(arr[i],``3``)`   `    ``determinant ``-``=` `3``*``arr[``0``]``*``arr[``1``]``*``arr[``2``]`   `    ``return` `determinant`   `# Driver code` `arr ``=` `[``4``,``5``,``3``]` `n ``=` `len``(arr)` `print``(calcDeterminant(arr,n))`   `# This code is contributed by Shrikant13`

## C#

 `// C# program for finding determinant` `// of generated matrix ` `using` `System;`   `class` `GFG` `{` `static` `int` `N = 3;`   `// Function to calculate determinant ` `static` `double` `calcDeterminant(``int` `[]arr) ` `{ ` `    ``double` `determinant = 0; `   `    ``for` `(``int` `i = 0; i < N; i++) ` `    ``{ ` `        ``determinant += Math.Pow(arr[i], 3); ` `    ``} `   `    ``determinant -= 3 * arr[0] * ` `                       ``arr[1] * arr[2]; `   `    ``return` `determinant; ` `} `   `// Driver code` `static` `public` `void` `Main ()` `{` `    ``int` `[]arr = { 4, 5, 3 }; ` `    ``Console.WriteLine(calcDeterminant(arr)); ` `}` `}`   `// This code is contributed by akt_mit`

## PHP

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## Javascript

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Output:

`36`

Time Complexity : O(N) as only one traversal is requires on array.

Auxiliary Space : O(1), since no extra space has been taken.

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