Suppose we are given a symmetric tridiagonal $4 \times 4$ matrix $A$ and perform QR factorization
$$A=QR$$
Then we define $A' := RQ$. Matrix $A'$ still possesses the tridiagonal structure where we can again repeat the steps to find the eigenvalues of $A$.
I know $A$ has to be symmetric tridiagonal (not just only tridiagonal), but is $A$ necessarily invertible? If so, please illustrate the proof or idea that invertibility is required. If not, please provide an example. Thank you very much.