The 18-year-old, who also lived in the Caerphilly area, admitted murder during a brief hearing at Newport Crown Court, which was attended by a large gathering of Mrs Williams' family.
The Recorder of Cardiff, Judge Nicholas Cooke QC, adjourned sentencing until Friday September 11. He said that if he could have dealt with the case now, he would have done, but it was not possible.
He said the sentence was fixed by law, but what the court has to determine is the minimum time for which he should be detained. In the meantime, Matthews was remanded in custody.
After the hearing, Detective Superintendent Geoff Ronayne, of Gwent Police, said it was still not clear what happened on the night of the murder due to Matthews' reluctance to talk about it.
He said the teenager had been living rough for a few nights after an argument with his girlfriend. He had been introduced to Mrs Jones, who the detective said had a "tremendous standing in the community."
"We know Matthews was taken to the address and she was likely to have put him up for the night and he repaid her by trying to steal property from the house and she disturbed him," said Mr Ronayne.
"We're still not sure why it happened but we do know that he carried out quite an horrific attack and he needed to be brought to book quite quickly."
He added: "I'm obviously extremely pleased for the family that they haven't had to go through the ordeal of a trial.
"It is the right result and I think they are pleased they are at the end of a harrowing experience and they can start to move on."
After the murder, Mrs Williams's family had paid tribute to her, calling her a "true carer" in the community.
During her career, she had worked in the maternity department of the Caerphilly Miners Hospital and as a district midwife.
At the time of Mrs Williams's death, her family said she was "much-loved" among fellow staff members and residents at the Bedwas home where she cared for the elderly and infirm.