n=3;q=Table[RandomReal[{-1,1}],{n},{n}];
m=Transpose[q].q;

minv=Inverse[m];Table[minv[[i,i]]=x[i],{i,1,n}];mtest=Inverse[minv];

eqs=Table[mtest[[i,i]]==m[[i,i]],{i,1,n}];

sol=NSolve[eqs,Table[x[i],{i,1,n}],WorkingPrecision->7];

list=Table[PositiveDefiniteMatrixQ[mtest/.sol[[i,All]]],{i,1,Dimensions[sol][[1]]}]

n=4;q=Table[RandomReal[{-1,1}],{n},{n}];
m=Transpose[q].q;

minv=Inverse[m];Table[minv[[i,i]]=x[i],{i,1,n}];mtest=Inverse[minv];

eqs=Table[mtest[[i,i]]==m[[i,i]],{i,1,n}];

sol=NSolve[eqs,Table[x[i],{i,1,n}],Reals];

list=Table[PositiveDefiniteMatrixQ[mtest/.sol[[i,All]]],{i,1,Dimensions[sol][[1]]}]

Table[{Round[x[1]/.sol[[i,1]],0.0001],list[[i]]},{i,1,Dimensions[sol][[1]]}]//DeleteDuplicates

n=5;q=Table[RandomReal[{-1,1}],{n},{n}];
m=Transpose[q].q;

minv=Inverse[m];Table[minv[[i,i]]=x[i],{i,1,n}];mtest=Inverse[minv];

eqs=Table[mtest[[i,i]]==m[[i,i]],{i,1,n}];

sol=NSolve[eqs,Table[x[i],{i,1,n}],Reals];

list=Table[PositiveDefiniteMatrixQ[mtest/.sol[[i,All]]],{i,1,Dimensions[sol][[1]]}]

Table[{Round[x[1]/.sol[[i,1]],0.0001],list[[i]]},{i,1,Dimensions[sol][[1]]}]//DeleteDuplicates

Dimensions[%]