Hundreds of mourners attended the funeral of the legendary centre at Llandaff Cathedral.
Morgan, a Cardiff and Wales team mate of Matthews, was unable to attend the funeral but sent a special tribute to the family which was read out by Canon Graham Holcombe.
"Dr Jack looked after me from my very first game for Cardiff when I was only 18. He was my first captain and it was a privilege to play with him and Bleddyn Williams," wrote Morgan.
"He became my inspiration. He then made the transition from a super rugby star to a much loved doctor who brought both my daughter, Catherine, and son, Nick, into this world.
"Dr Jack wanted to join the Air Force during WW2, but was ordered to continue with his medical studies. They needed more doctors than pilots. He once told me that 85% of his the pilot intake he had hoped to be among had died in training.
"Our generation, he used to say, the ones who survived or who were spared, have spent the rest of our lives trying to make every last minute count one way or another. It was the deal we struck privately with ourselves to keep our sanity and to honour those who didn't make it. That's what it has all been about.
"Many people walk in and out of your life, but only true friends leave footprints in your heart. Dr Jack will live in my heart forever."
Dr Jack, as he was universally know, was Wales' oldest international at the time of his death, passed away in his home town of Bridgend last week aged 92. He made 20 appearances for the Lions and later became the team's doctor.
Rees was joined by a host of other former Welsh internationals and Lions such as, John Dawes, Barry John, JPR Williams, Gerald Davies, D Ken Jones, Rob Norster, John O'Shea, Rob Howley, Alan Phillips and current cap Jamie Roberts.
Former Wales hooker Brian Rees, an eminent surgeon, gave an appraisal of Dr Jack's 50 year career as a medical GP.
"His commitment to both rugby and medicine was equal. When he went on the 1950 Lions tour he had to pay for a locum to take over at his Practice to work in his place while he was battle the All Blacks," said Rees.
"His life would fill the script of a Hollywood movie, but then Dr Jack wouldn't have liked that. He gave freely of himself to Cardiff and to Wales and made friends wherever he went.
"He carried on as a GP well after the retirement age and was welcomed into people's homes as an extension of their family. One of his proudest boasts to me was not about rugby, but the fact he had never lost and hurt a baby in delivery throughout his long career."