ANZ's standard variable rate for owner-occupier principal-and-interest loans will fall to 5.18 per cent, with the interest-only rate coming down to 5.91 per cent.
In the case of Westpac, the treasurer said the bank owes its customers an explanation for keeping some of the RBA's rate cut to themselves.
The Reserve Bank of Australia has cut interest rates bringing it to a historic low of 1.25 per cent.
While ANZ responded to the RBA's 0.25 percentage point cut with a 0.18 basis point reduction to its variable rate home loans, Commonwealth and NAB said they would go further as they rebuild amid the fallout of last year's financial services royal commission.
"The combination of the tax cuts and today's RBA decision could see a two-income family, for example a teacher and a tradie, each earning $60,000 a year, with a $400,000 mortgage nearly $3,000 a year better off as a household", Frydenberg said.
"The Board took this decision to support employment growth and provide greater confidence that inflation will be consistent with the medium-term target".
APAC economist at Indeed.com, Callum Pickering, said consistently low inflation had ultimately forced the RBA's hand. "It is a sign of deep underlying concerns across the Australian economy", Pickering said.
"My message to the banks is that, while they are critical pillars of the economy and especially at a time of domestic and global economic challenges, it was important to maintain the flow of credit to households and businesses", Frydenberg told The Australian on Tuesday.
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"Given this, the possibility of lower interest rates remains on the table".
He told reporters in Brisbane, "If only this government spent less time bagging Labor or pointing the finger at others and more time focused on the substantial weaknesses in the economy it wouldn't be floundering as badly as it is today".
Following ANZ's announcement, Frydenberg said the bank had "let down" its customers with its decision not to pass the cut on in full.
About an hour later, CBA said it would pass on the full RBA cut - albeit only from June 25.
The bank's net interest margin - the difference between interest charged on a loan and that paid to fund it - has been squeezed in recent times.