About decomposition of singular values and singular systems

Question

Consider the example from the book Robust control and filtering of singular systems (Please see attached image example). I'm trying to reproduce the same example using Matlab. The problem is when I'm using the svd function I got the matrices N and M which does not satisfy the property MAN equals to the diagonal of A and one. So to specify the soloution among all the possible SVD decompositions. Thank you. Example in the book:

A=[-10 5 6.5
    2 -5.5 -1.25
    -9 4 8.5];
E=[1 1 0.5
    -0.5 1.5 1.75
    1 1 0.5]; M=[2 -1 1
    1 0.5 -1.5
    2 -1 0];
N=[0 1 1
    -1 1 1.5
    -1 1 -2]. 

Here is my Matlab code which permits to find matrices M and N such that M[I 0 0;0 I 0;0 0 0]N=E but it does not guarantee the second statement "inv(M)Ainv(N)=[A_bar 0;0 I]"

n=3;
[M,I2,N] = svd(E);
E1=rref([I2 inv(M)]);
I2= E1(:,1:n); % the reduced row echelon form of c
M= inv(E1(:,n+1:end)); % now we have T
N=N';