Let's say I have an inverting amp with the + input at ground. The feedback resistor Rf is 100K, and the input resistor R1 to the inverting input is 10K. That's a gain of -10.
Now suppose I add R2, a 10K resistor from the inverting input to ground.
It seems this would cut the input signal in half, and lower the input impedance, but doesn't change the gain. I say that because at virtual ground at the inverting input, there is no voltage difference across R2, and therefore no current will flow through it. Hence no effect on the gain.
And yet, somehow that doesn't seem right. Well, of course I could breadboard it and see, but what's the fun in that?