Yes, but it will require the use of VBA. That means everyone who uses the workbook will have to enable macros. Are there any other macros already in this workbook? If not, then this would be a notable change. If so, then it's not big deal.
Here's a macro that would do it:
How to use the macro:
1. Open up your workbook
2. Right-click on the sheet tab
3. Select VIEW CODE from the popup context menu
4. Copy and Paste in your code (given above)
5. Get out of VBA (Press Alt+Q)
6. Save your sheet
The macro is installed and ready to use.
Now make changes to cell A1 and watch what happens.
You will have to save the workbook as a macro-enabled (*.xlsm) file.