QR decomposition in Eigen C++ - sign problem - c++

Question

I need to compute some determinants for a project: I use c++ 14 and Eigen.
So, MatrixXd A is a Eigen matrix with X rows and X cols and contains double values. To compute determinant I use A.determinant(). Let's pretend that A.determinant() is equal to d. Then, the problem apper when I use QR decomposition because R.determinant() is equal to -d, should be equal to d. This happened only for large matrices (with size greater than 5 - I observed this). Only the sign is problem, why?
My code:
#include <iostream>
#include <Eigen>
#include <fstream>
#include <chrono>
using namespace Eigen;
using namespace std;
using namespace std::chrono;
ifstream fin("input.txt");
int main()
{
double aux;
int n = 10;
MatrixXd A;
A.resize(n,n);
// Read A
for(int i=0;i<n;i++)
for(int j=0;j<n;j++){
fin>>aux;
A(i,j) = aux;
}
cout<<"Start!"<<endl;
cout<<A.determinant()<<endl;
//Use QR decomposition, get R matrix
HouseholderQR<MatrixXd> qr(A);
qr.compute(A);
MatrixXd R = qr.matrixQR().template triangularView<Upper>();
// R is a triangular matrix, det(A) should be equal to det(R)
cout<<R.determinant()<<endl;
return 0;
}
How can I solve this?
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