The 42-year-old was confirmed as Charles Green's temporary successor in a statement to the London Stock Exchange on Wednesday and also takes on the position of chief operating officer. Mather is a shareholder with a 3.1 per cent stake in Rangers and has been the club's director of sports development since October.
Murray said: "Craig's appointment as chief operating officer will allow the company and the club to keep moving forward. Craig has invested £1million of his own money in the club and is committed to playing an active part in delivering future success for Rangers."
He added: "The board has approved his appointment and look forward to him pushing us in the correct direction."
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.
Murray added: "This process will be completed as swiftly as possible and no-one should jump to conclusions regarding the outcome. 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. Now is the time to move forward and the most important thing for all Rangers fans, myself included, is ensuring our club can play as constructive a part in Scottish football as possible."
Mather - who wants the CEO job on a permanent basis - is determined to dispel any doubts fans may have about his appointment. He acknowledges that some fans may be wary of him following another turbulent period off the field but has vowed to have a positive impact on the club.
He said: "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. I hope to dispel all of those if that is the case. They deserve so much better than they've been getting of late."