Over the last week, Opposition MPs had raised concerns about whether Matthews failed to pick up or act on fraudulent activity in his former role as Secretary of Transport.
Matthews led the transport ministry while former senior manager Joanne Harrison stole $750,000 through fake invoices. She was jailed for three years and four months in February.
Earlier this week, Carter stood by Matthews, saying the Serious Fraud Office had described his performance as exemplary during the fraud investigation into Harrison.
Carter today said that he still had complete confidence in Matthews.
"No information has been presented to me that leaves me to doubt that confidence," he said.
That position was echoed by Assistant Speaker and Labour MP Trevor Mallard.
"We're not going to hang the Auditor-General without a proper look," he said.
Matthews is answerable to Parliament, not Government, and his appointment earlier this year was supported by all parties.
The appointment process took the extraordinary step of hearing from the Serious Fraud Office about Matthews' handling of the Harrison case.
However, the SFO's briefing was limited to events after April 2016, when the agency opened its case into Harrison.
Documents released since Harrison was convicted show that ministry staff had alerted Matthews to Harrison's unusual activities as early as 2013.
Sir Maarten's inquiry will dig deeper into the case. He will "go back as far as he needs to" to investigate Matthews' handling of the Harrison affair, Carter said.
In a statement, Matthews said he stood by his actions, which were based on the information available to him at the time.
"However, the current media coverage about these matters has the potential to undermine the important constitutional role of the Controller and Auditor-General."
Deputy Auditor-General Greg Schollum will take over while Matthews is being investigated.
Earlier today, the State Services Commission announced it was taking over an inquiry into whether staff at the ministry lost their jobs after raising concerns about Harrison's behaviour.
That inquiry had previously been led by the transport ministry itself, but its handling of the investigation had been criticised by former ministry staff.