h := 1 /* Initialization of the pivot row */
 k := 1 /* Initialization of the pivot column */
 while h ≤ m and k ≤ n
   /* Find the k-th pivot: */
   i_max := argmax (i = h ... m, abs(A[i, k]))
   if A[i_max, k] = 0
     /* No pivot in this column, pass to next column */
     k := k+1
   else
      swap rows(h, i_max)
      /* Do for all rows below pivot: */
      for i = h + 1 ... m:
         f := A[i, k] / A[h, k]
         /* Fill with zeros the lower part of pivot column: */
         A[i, k]  := 0
         /* Do for all remaining elements in current row: */
         for j = k + 1 ... n:
            A[i, j] := A[i, j] - A[h, j] * f
      /* Increase pivot row and column */
      h := h+1
      k := k+1

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>

typedef std::vector<std::vector<int>> matrix;
typedef long long ll;

void inverse_matrix(matrix &mat)
{
    ll h = 1, k =1;
    auto m = mat.size(), n = mat[0].size();


    while (h <= m && k <= n)
    {
        ll i_max = 0;
        for (ll i = h; i <= m; ++i)
        {
            i_max = std::fmax(i, std::abs(mat[i][k]));
        }

        if (mat[i_max][k] == 0)
        {
            ++k;
        }

        auto temp = mat[h];
        mat[h] = mat[i_max];
        mat[i_max] = temp;

        for (auto j = h + 1; j <= m; ++j)
        {
            auto f = mat[j][k] / mat[h][k];
            mat[j][k] = 0;

            for (auto v = k + 1; v <= n; ++v)
            {
                mat[j][v] = mat[j][v] - mat[h][j] * f;
            }
        }

        ++h;
        ++k;
    }
}

int main() {
    matrix mat = {{2, 2}, {4, 5}};
    inverse_matrix(mat);

    return 0;
}