This means that you can verify in MATLAB similarity of matrices by checking with the command rref that they have the same reduced row echelon form.

>> A = [4 -2; 2 1]
A =
     4    -2
     2     1
>> B = [3 -2; 1 2]
B =
     3    -2
     1     2
>> rref(A) == rref(B)
ans =
  2×2 logical array
   1   1
   1   1
>> isequal(rref(A), rref(B))
ans =
  logical
   1
>> T = [1 -1; 1 0]   % appropriate transformation matrix
T =
     1    -1
     1     0
>> T^(-1) * A * T    $ equal to matrix B
ans =
     3    -2
     1     2
>> B  -  T^(-1) * A * T
ans =
     0     0
     0     0
>> S = [1 1; 0 2]   % transformation matrices are not unique
S =
     1     1
     0     2
>> B - S^(-1) * A * S
ans =
     0     0
     0     0
>> P = S * T^(-1)   % non-trivial transformation matrix that leaves A intact
P =
    -1     2
    -2     2
>> A - P^(-1) * A * P  
ans =
     0     0
     0     0