I have typed this code to perform the LU factorization for A=xb when we are given matrix A and $x=(1,1,1)^T$
. My code is below

function q1()
    % Define x_ex
    x_ex = [1; 1; 1];

   % Define epsilon for given k
    for k = 0:9
        epsilon = 10^(-k);
        
        % Define matrix A
        A = [2, -2, 0;epsilon - 2, 2, 0;0, -1, 3];
        
        % formula to find vector b
        b = A * x_ex;

         % LU factorization
        [L, U, P] = lu(A);
       
        y = L  (P * b);
        
        x_ex = U  y;

        fprintf('For epsilon = 1e-%d:n', k);
        fprintf('Vector b:n');
        disp(b');
        fprintf('For k=%d (epsilon=10^-%d):n', k, k);
        disp('Permutation matrix P:');
        disp(P);
    end
end

However when I run this code P should return the identity and instead I am getting P = [1,0,0;0,0,1;0,1,0]. Is there an issue in my code that I am missing?