I have noisy pipes in a house built in 2007 - banging when we turn off the hot water in an upstairs sink and loud 'bang bang bang' when the washer switches from cold to hot. We have a hot water circulating pump and two water heaters in the garage.

Upstairs on the sink, when you turn off the hot water there is a banging above the adjacent shower. Pictures taken during construction show no known pipes above this 2nd story bathroom - only attic. The plumber said that it is echoing through the space between the studs inside the wall. (?) They opened up the sheetrock in a 1st story bathroom directly below it and braced the pipes - no change. We think it is air that is trapped in the pipes.

We have (several times) cut off the water at the street and opened all faucets - this sometimes softens the noise but does not eliminate the problem.

The only answer the builder (who has been good to us) and the plumber have come up with is to change out the cartridges in the 4 showers in the house. I don't see how this would help when the noise happens when we turn off the sink?

Can anyone explain this or give me any other solutions? Does anyone think there would be pipes over a second floor shower that would cause the banging and we should go up in there?

Thank you!

change out the cartridges in the 4 showers in the house.
That certainly won't help.

What you are experiencing is called "water hammer". When any fast acting valve such as a toilet, washer or a a single lever faucet is closed, the water flow is abruptly stopped. This causes the pipes to bang against the structural member to which they are attached.
If the person who did the plumbing was a qualified plumber he installed air chamber in the walls at each fast acting valve. An air chamber is just a piece of vertical pipe that is capped at the top. When the water is turned on, air in the pipes rises up into these air chambers. When the flow of water is suddenly stopped and the air in the air chambers act like shock absorbers. Unfortunately over time the air in the air chambers is absorbed by the water and they become ineffective.
To restore the air in the air chambers, turn off the water and open all faucets and flush all toilets. This will of course drain the water from all the pipes. Again unfortunately, the water in the air chamber can be held in by a vacuum. So give it some time, eventually the water will break the vacuum and the air chambers will drain also.
You then close all faucets and turn on the water. The air in the pipes will naturally rise to the air chambers. Probably also help if you can wait a while to open faucets.

There is also something called water hammer arrestors. Same thing air chambers but are usually only 6 to 8" high.
See image below.
They have a rubber membrane in side that keeps the air and water separated. They can be installed any where along an exposed pipe. Because they have the rubber membrane inside they do not have to be installed vertically. Can be installed horizontally or even upside down.

Try draining you pipe again, but do it when you can wait a period of time before turning the water back on. If you have any exposed piping consider installing some water hammer arrestors.

Replacing shower cartridges will not help.