The transaction was successfully authorised with the bank
can I substitute with in the sentence with by? If so, does by the bank have any different meaning than with the bank?
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A transaction involves at least two parties. Let us assume that they are a bank and a customer.
To say that the transaction was authorised by the bank means that the bank approved the transaction with the customer. The customer presumably wished to borrow money and the bank gave its approval to the loan.
To say that the transaction was authorised with the bank is somewhat ambiguous. In this instance it is not clear who is doing the authorising. It is possible that some third party, eg. a government body that oversees fair competition, gives the green light to a deal between the bank and the customer. Where company take-overs and mergers are concerned, this is frequently the case.
What is certain is that in this context the two prepositions are not interchangeable.
By the bank signifies the act of authorisation rested solely with the bank and potentially implies you were not involved with the transaction.
With the bank is functionally the same in common usage, but it implies you were involved/aware of the transaction.
Day to day both versions would likely be used interchangeably because the dominant act, that of authorisation, rests with the bank in both cases.