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

> library(pracma)
> A <- matrix(c(4, 2, -2, 1), nrow=2); A
     [,1] [,2]
[1,]    4   -2
[2,]    2    1
> B <- matrix(c(3, 1, -2, 2), nrow=2); B
     [,1] [,2]
[1,]    3   -2
[2,]    1    2
> rref(A) == rref(B)
     [,1] [,2]
[1,] TRUE TRUE
[2,] TRUE TRUE
> all.equal(rref(A), rref(B))
[1] TRUE
> T <- matrix(c(1, 1, -1, 0), nrow=2); T  # appropriate transformation matrix
     [,1] [,2]
[1,]    1   -1
[2,]    1    0
> inv(T) %*% A %*% T   # T^(-1)*A*T is equal to matrix B
     [,1] [,2]
[1,]    3   -2
[2,]    1    2
> S <- matrix(c(1, 0, 1, 2), nrow=2); S  # transformation matrices are not unique
     [,1] [,2]
[1,]    1    1
[2,]    0    2
> B - inv(S) %*% A %*% S
     [,1] [,2]
[1,]    0    0
[2,]    0    0
> P <- S %*% inv(T);  P  # nontrivial transformation matrix that leaves A intact
     [,1] [,2]
[1,]   -1    2
[2,]   -2    2
> A - inv(P) %*% A %*% P
     [,1] [,2]
[1,]    0    0
[2,]    0    0