Yes, you can. The key concept is Linked Data. You can access this menu with Ctrl+L or CMD+L on Mac OS.

First select all your duplicates. Then select the origin, or what you want to be the origin, last. It is important you select it last, this makes it the active object. You can see this because it is outlined in yellow instead of orange. Press Ctrl+L and select Link Object Data.

In this screenshot I have duplicated two cubes from the original, and rotated them so you can see that linking them does not alter their location or rotation. The Link/Transfer Data menu is active.

Once they are linked, if you make changes to the original object in Edit mode, or scaling e.g., it will affect all linked duplicates:

Yes, you can. The key concept is Linked Data. You can access this menu with Ctrl+L or CMD+L on Mac OS.

First select all your duplicates. Then select the origin, or what you want to be the origin, last. It is important you select it last, this makes it the active object. You can see this because it is outlined in yellow instead of orange. Press Ctrl+L and select Link Object Data.

In this screenshot I have duplicated two cubes from the original, and rotated them so you can see that linking them does not alter their location or rotation. The Link/Transfer Data menu is active.

Once they are linked, if you make changes to the original object in Edit mode it will affect all linked duplicates: