I am guessing the condition number of your matrix.
The error in matrix inversion get's worse as the condition number of a matrix increases. When your matrix is 10x10 the condition number of your specific matrix is around 4. while when your matrix is 200x200 it's condition number is around 90. So this means that small perturbation will be amplified until the error will dominate the solution.
Although I have to say that these condition number are not that high. Actually they are small enough. But anyway, one thing that you can use is using some sort of conditioner matrix. So, instead of solving Ax=b you would solve PAX=Pb, where P is your conditioner which hopefully (PA) has better condition number.
before going to use conditioner, just make sure that you are constructing your A matrix properly and there isn't code error.
something like this should do it:
A=-tril(ones(n,n),-1)+eye(n); A(:,end)=1;
By the way, in your case, it appears that you can analytically invert these matrices. The inverted matrix can be constructed as follow:
n=200;
Ainv=eye(n,n)/2;
for i=1:n-1
Ainv(i, (i+1:n-1) )= -2.^-(2:(n-i));
end
Ainv(n,1:n-1)=2.^-(1:n-1);
Ainv(1:n-1,n)=Ainv(1:n-1,n-1);
Ainv(n-1,n)=-0.5;
Ainv(n,n)=Ainv(n,n-1);
