void gem(matrix &matrix, int level){
    if(level == matrix.size() - 1)
        return;
    auto divider = matrix[level][level];
    auto row = level;
    // find non zero divider
    while (divider == 0 && row < matrix.size()){
        divider = matrix[++row][level];
    }

    if(divider == 0){
        gem(matrix,level + 1);
        return;
    }

    // divide row
    for (int i = 0; i < matrix[row].size(); ++i)
        matrix[row][i] /= divider;

    // elimination
    for (int i = row + 1; i < matrix.size(); ++i) {
        auto scaler = matrix[i][level];
        for (int j = level; j < matrix[i].size(); ++j) {
            matrix[i][j] -= scaler * matrix[row][j];
        }
    }
    gem(matrix,level + 1);
}