> For the complete documentation index, see [llms.txt](https://gl01.gitbook.io/gfg-editorials/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://gl01.gitbook.io/gfg-editorials/2023/12-2023-dec-24/25-determinant-of-a-matrix.md).

# 25. Determinant of a Matrix

The problem can be found at the following link: [Question Link](https://www.geeksforgeeks.org/problems/determinant-of-a-matrix-1587115620/1)

!\[]\(<https://badgen.net/badge/Level/So> Call Easy/green)

## My Approach

To calculate the determinant of a matrix, I have implemented the following steps:

1. If the matrix is 1x1, return the only element (`mat[0][0]`)
2. If the matrix is 2x2, return the determinant using the standard formula (`mat[0][0]*mat[1][1]-mat[0][1]*mat[1][0]`)
3. For larger matrices, use a recursive approach:
   * Create a submatrix for each element in the first row.
   * Recursively calculate the determinant of each submatrix.
   * Sum the products of each element in the first row, its corresponding submatrix determinant, and a `sign` factor.
   * The sign factor alternates between 1 and -1.

## Time and Auxiliary Space Complexity

* **Time Complexity**: `O(n!)`, factorial time complexity due to recursive calls
* **Auxiliary Space Complexity**: `O(n^2)`, space for the submatrix

## Code (C++)

```cpp
class Solution {
public:
    int determinantOfMatrix(vector<vector<int>> mat, int n) {
        if (n == 1) {
            return mat[0][0];
        }
        if (n == 2) {
            return mat[0][0] * mat[1][1] - mat[0][1] * mat[1][0];
        }

        int sign = 1, ans = 0;
        vector<vector<int>> m(n - 1, vector<int>(n - 1));

        for (int row = 0; row < n; ++row) {
            int k = 0;
            for (int i = 0; i < n; ++i) {
                if (i != row) {
                    for (int j = 1; j < n; ++j)
                        m[k][j - 1] = mat[i][j];

                    k++;
                }
            }
            ans += sign * (mat[row][0]) * determinantOfMatrix(m, n - 1);
            sign *= -1;
        }
        return ans;
    }
};
```

## Contribution and Support

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```

Please replace the placeholders like `step 1: explanation` with your actual explanation of the approach.
```