find inverse matrix using naive gaussian elimination

Hey guys, I've been working on this assignment I found online. I have to extend my naive gaussian elimination code to find the inverse matrix. I have the nge code but don't know how to go on from there and use it to find the inverse of say M = [1 2 3; 4 5 6; 7 8 1], I know I need [M | I] and make M into I and the right hand side would be M^-1. Thanks.

if true
  % function x = naive_gauss_elim(A,b)
n = length(b);
x = zeros(n,1);
% Forward elimination to get upper triangular matrix
for k = 1:n-1
  for i = k+1:n
      xelem = A(i,k)/A(k,k);
      for j = k+1:n
          A(i,j) = A(i,j)-xelem*A(k,j);
      end
      b(i) = b(i)-xelem*b(k);
  end
end
% back sub
x(n) = b(n)/A(n,n);
for i = n-1:-1:1
  s = b(i);
  for j = i+1:n
      s = s-A(i,j)*x(j);
  end
  x(i) = s/A(i,i);
end

end

[Andrei Bobrov]

M\eye(size(M))

Answer 1

[Andrei Bobrov]

[m,n] = size(M);
A = [M, eye([m, n])];
for k = 1:m
    A(k,k:end) = A(k,k:end)/A(k,k);
    A([1:k-1,k+1:end],k:end) = ...
                    A([1:k-1,k+1:end],k:end) - A([1:k-1,k+1:end],k)*A(k,k:end);
end
out = A(:,n+1:end);