During his first Question Time as Prime Minister, Malcolm Turnbull has been forced to defend the Coalition's economic performance and its stances on same-sex marriage and climate change.

The day after the Liberal party room split 54 to 44 in his favour, Mr Turnbull on Tuesday set out to heal the wounds of the leadership challenge by praising the man he deposed, Tony Abbott, as a "great Australian".

The nation's 29th Prime Minister also met the Nationals on Tuesday morning to strike a fresh bargain for Coalition government.

Mr Turnbull will spend the next few days assembling a new cabinet and a new management team for the prime ministerial office, pledging to run a more collegiate, cabinet-focused government.

Key points: Malcolm Turnbull is sworn into office

Malcolm Turnbull is sworn into office Faces his first Question Time as Prime Minister

Faces his first Question Time as Prime Minister Calls the man he ousted a "great Australian"

Calls the man he ousted a "great Australian" Says "we owe a great debt to Tony Abbott"

Says "we owe a great debt to Tony Abbott" New Cabinet line-up not expected until weekend

About an hour before Mr Turnbull was sworn in at Government House, Mr Abbott delivered his own speech, conceding he had been ousted as a first-term prime minister after just under two years in office.

The former leader partly blamed internal "white-anting" and "character assassination" in the media for his demise but promised not to "wreck" his Government in retribution.

On Monday Mr Turnbull justified his decision to challenge for the leadership by criticising the Government's economic record and its policy sales job.

Mr Abbott did not turn up to his allocated backbench seat during Question Time, where Mr Turnbull began by arguing his predecessor should be recognised for his achievements.

"Our nation, our parliament, our government, our party — our parties, the Coalition — owe Tony Abbott an enormous debt of gratitude for his leadership and his service over many, many years," he said.

"He led us out of opposition and back into government.

"Tony has discharged his role as prime minister — be it as leader of the opposition — with enormous distinction and achievement."

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That comment was met with laughter from the Labor benches, but the new Prime Minister pressed on.

"We owe a great debt to Tony Abbott, we thank him for his leadership, we thank him for his service, he is a great Australian and our country has been better, has been improved has been better led under his time as prime minister," Mr Turnbull said.

Mr Turnbull said the free trade agreements clinched by the Abbott government were an example of the success.

He also congratulated Mr Abbott for his border protection policies, arguing they allowed the government to provide for an additional 12,000 Syrian and Iraqi refugees.

Former prime minister John Howard congratulated Mr Turnbull and praised him for his "great intelligence".

"He has the capacity to explain economic concepts very clearly and very lucidly and that ... will be an important part of the skill set that he brings to his new responsibilities," Mr Howard told reporters in Sydney.

He spoke to Mr Turnbull on Tuesday and offered him guidance.

"He will have all the support and advice that he may care to seek from me. As much or as little as he chooses," Mr Howard said.

Turnbull commits to 'people's vote' on same-sex marriage

In Question Time, Mr Turnbull confirmed the Coalition would stick to its policy to hold a "people's vote" on same-sex marriage after the next election.

"There is no greater virtue in a free vote here or a plebiscite, they are each means of resolving the matter," he said.

"One I grant you is more expensive but, nonetheless, it is a very legitimate and democratic way of dealing with it."

Mr Turnbull rejected Labor's taunts that he was a "sell-out" on the issue.

Malcolm Turnbull is sworn in as Prime Minister alongside deputy Liberal leader Julie Bishop at Government House.

"Why is the Opposition afraid of the people having a vote?' he said.

"Why don't they want all Australians having a vote?"

Under questioning from the Greens, the Prime Minister also confirmed he is sticking with the Coalition's climate change policy.

"There are many means of cutting emissions, some more complex than others, some more expensive than others," Mr Turnbull said.

"There is no inherent virtue in any particular method of cutting emissions. The targets we have set are reasonable ones."

Shadow treasurer Chris Bowen mocked comments Federal Treasurer Joe Hockey made on Monday.

"Does the Treasurer stand by his statement of yesterday in relation to the now Prime Minister?" Mr Bowen said.

"And I quote, 'he's never said to me or the Cabinet that we are heading in the wrong economic direction'."

Mr Hockey accused Mr Bowen of verballing him, despite on Monday saying: "He [Mr Turnbull] has never said to me, or to the Cabinet, that we are heading in the wrong economic direction."

New Cabinet line-up not expected until weekend

The make-up of Mr Turnbull's Cabinet it not expected to be known until the weekend, or perhaps even Monday, according to a Coalition briefing.

Mr Turnbull told Tuesday morning's party room he is committed to a "broad church" approach to governing and is expected to make significant changes to the ministry.

Social Services Minister Scott Morrison looks likely to be handed the treasury portfolio, replacing Mr Hockey, who has been under fire from within his own party virtually ever since his first budget.

Several of Mr Turnbull's supporters believe elevating Mr Morrison to the role of economic salesman is key to the stability and function of the Government.

There is speculation Education Minister Christopher Pyne will become defence minister, where he could oversee the process to acquire Australia's next fleet of submarines.

South Australian MPs and senators, who feared electoral wipe-out with Mr Abbott at the helm, played a significant role in the leadership change and many want the submarines to be built locally in Adelaide.

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Attorney-General George Brandis says he wants to keep his job, becoming the first Cabinet minister to reveal that he voted for Mr Turnbull in the leadership spill.

He told the ABC's 7.30 program that he and his colleagues had been growing increasingly concerned about the Coalition's re-election chances.

"We were at real risk of losing next year's election to Bill Shorten. That would be a catastrophe for Australia, and it was a very difficult decision for all of us who've served with Mr Abbott and respect him."

There is also a push for Mr Turnbull to promote more women and thrust younger talented Liberals, who currently hold junior roles, into the limelight.

There is a long list of contenders likely to put their name forward for higher honours, but South Australian senator Simon Birmingham, Victorian Kelly O'Dwyer and West Australian Christian Porter are among those often mentioned.

It is expected some senior Cabinet supporters of Mr Abbott, including Kevin Andrews, could be dumped in the shake-up.