Letters of Administration will show that, for example, Fred Bloggs of 23 Acacia Avenue, Sometown has the legal right to administer the estate of the deceased but Fred Bloggs would still need to prove to the deceased person's bank that he really is the said Fred Bloggs and not someone (who happens to have found the Letters of Administration left on a train) simply pretending to be Fred Bloggs.
So, while physical possession of Letters of Administration might help Hardy49 establish her identity with her brother's bank, it's still not impossible that they could want to see further proof of identity before they'll release the funds to her. It really comes down to an individual bank's security protocols, rather than to a precise matter of law, in my opinion.