The 42-year-old, who is a shareholder and has been the club's director of
sports development since October, was confirmed as Charles Green's temporary successor on Wednesday morning.
Green resigned with immediate effect on Friday amid allegations by former owner Craig Whyte that he was involved in the Green consortium's acquisition of Rangers last summer.
The Yorkshireman has denied any wrongdoing, while the Rangers board has commissioned an independent investigation into the claims by Whyte, who was in charge when the club went into administration last February.
"Rangers fans have been outstanding in their support for the club during the most difficult of times and I know they don't know me and may have some doubts," Mather said.
"I hope to dispel all of those if that is the case. I know there have been rumours about who is supposed to be connected with whom but I prefer to distance myself from all of that.
"I know what my own hard work has achieved over the years and where it has taken me and I will use of all of my experience and knowledge to help improve Rangers.
"I could not stand here and tell you I have always been a Rangers fan but I can say, with profound honesty, that I have been around football all my life and I have known of Rangers and their huge successes for many years. But then who hasn't?
"I am well aware of what this club means to so many people. They deserve so much better than they've been getting of late. My job will be to make sure the work is put in to make them even prouder to be Rangers fans.
"I look forward to starting work immediately and would like to make it clear the chief executive's job is a position I would like to take on permanently."
Rangers chairman Malcolm Murray welcomed the appointment, saying: "We are very pleased to announce the appointment of Craig Mather as Chief Operating Officer of Rangers.
"He will also be acting Chief Executive which will allow us some time to examine the credentials of candidates for this position on a permanent basis.
"Craig has invested £1m of his own money in the Club and is committed to playing an active part in delivering future success for Rangers. The Board has approved his appointment and look forward to him pushing us in the correct direction.
"I would also like to assure all of the supporters that this money is being used for the benefit of the Club and nothing or no one else."
On the investigation Murray added: "The Board has also instructed the law firm Pinsent Masons to conduct the investigation into alleged links between Craig Whyte and former and current personnel at the Club.
"The fans deserve to be reassured on clarity, transparency and asset ownership and this is the main purpose of the inquiry.
"The last week or so has been turbulent for the Club but we should bear in
mind that the process of rebuilding is well under way."