The council sought for months to withhold the judges' scorecards from the January design contest to reshape the square.
But last week, Scotland's Freedom of Information (FoI) watchdog ordered the cards' release - and they show Matheson gave top marks to his pet project and much lower scores to the five other options.
Minutes after his choice was rejected by his fellow judges, Matheson dramatically announced the £15 million overhaul of the square was cancelled. The aborted exercise cost taxpayers £100,000 and the architects involved £200,000.
Matheson, who put the U-turn down to public opinion, was accused of "throwing his toys out the pram" because he didn't get his own way.
The council leader's behaviour is being investigated by the ethics watchdog for Scotland's councillors, after the body which ran the design competition, the Royal Incorporation of Architects in Scotland (RIAS), lodged an official complaint.
The RIAS claimed that, after insisting on chairing the panel, Matheson said before judging began he wanted a scheme by architect Burns & Nice to win.
When a design by John McAslan & Partners was named winner the scheme was scrapped. The square is now getting a "facelift" instead.
With Matheson under fire, the Sunday Herald used FoI legislation to ask for the scorecards in January.
The council repeatedly refused to release them, saying it would violate the "private thinking space" of the judges, harming future projects.
But, after an appeal, the Scottish Information Commissioner last week ordered full disclosure.
Matheson's scorecard is messy with corrections. He gave a maximum 90 points to his preferred option. The McAslan design got 30, the other four designs 30 to 60 points. Glasgow's SNP opposition leader Graeme Hendry said: "As the only scorecard with repeated changes, it certainly looks like [Matheson] either didn't know what he was doing or was trying to skew the result."
The RIAS said that faced with documentation from the case, the Public Standards Commissioner would understand how seriously the Councillor's Code had been breached.
A council spokesman said: "The scoring sheets comprehensively disprove the part of the RIAS allegation which relates to the scores. Contrary to RIAS's allegation, Cllr Matheson gave no zero scores."