Mather had been steering the club's finances and transfer activity on an interim basis since April, when Charles Green resigned following a controversial stint and amid a club-commissioned investigation into links with former owner Craig Whyte.
The boardroom has since seen more changes following the departure of Green and ally Imran Ahmad, with Malcolm Murray replaced by Walter Smith as chairman before leaving the board along with Phil Cartmell earlier this week, as Greenock-based businessman James Easdale moved in.
Murray, whose position had come under threat in May when a number of shareholders launched a formal attempt to oust both him and Cartmell, delivered a parting warning in an open letter to fans.
Murray expressed concern over a potential lack of transparency and the effect of the departures on the confidence of investors.
Mather today sought to assure supporters that matters were calm at Ibrox with former manager Smith playing an active role.
In an interview with Rangers TV, the English businessman said: "There is a lot of talk, of gossip, but the truth of the matter is that there is stability within the club, within the board.
"Walter has been fantastic for me, a great sounding board and I have to take my hat off to him because he went to America on holiday for three weeks and he rang me every day.
"He wanted to make sure everything was okay and was moving as we had planned.
"It's a job that I wanted and am fortunate enough to now have on a permanent basis and I intend to do something with that, to drive the club forward with stability."
Mather invested £1million in Rangers shortly after Green's consortium bought the assets and business of the liquidation-bound club for £5.5million in June last year.
And he claimed he would have still played an active role at Ibrox if his appointment as chief executive had not been ratified.
"I said right at the beginning that there was no way I would walk away and one of the reasons we came up with the structure that we did with me acting as chief operating officer and interim CEO was to give stability," he said.
"If a different candidate had been lucky enough to get the role, I would still been there to offer support based on the job I had been doing.
"Everything has been about the stability of the club and it's important now that everyone realises that and lets us get on with the business as usual."
Interim results showed Rangers lost £7million in their first six months of trading as a new company but that was offset by a £22million share issue.
The Third Division champions' summer transfer activity has raised some eyebrows with manager Ally McCoist still looking to sign defenders after bringing in seven players.
But Mather believes he can manage what he called the "balancing act" of providing McCoist with a team able to drive Rangers to the top flight while ensuring financial security.
"Rest assured I will be looking after the pennies and the pounds as well as hopefully being able to create a strategic path to get us back to the pinnacle of Scottish football and eventually Europe," the 42-year-old added.
"The additions were calculated additions, not knee-jerk reactions to anything.
"I was extremely keen to get the new faces in as early as possible, again for stability and the impetus going into the first game.
"Doing what we've done, I'm hoping we can start off with a bang."
McCoist is keen on Bosnia defender Boris Pandza and French-Tunisian centre-back Bilel Mohsni. The latter will remain with the squad for their second pre-season tour after impressing in a two-game stint in the Highlands.
The former Southend player will be joined by one of McCoist's summer signings, Arnold Peralta, who has flown in from Honduras after being given extra time off following international duty.
The midfielder has shaken off a hamstring injury to be fit to join Rangers as they set off for Germany via a friendly against Bristol City at Ashton Gate tomorrow.
Winger David Templeton will stay behind with what the club described as a "minor knock".