Noise amplification with inverse filtering
The noise amplification would be the ratio of the norms:

$\frac{\text{Norm of noise after inverse filter}}{\text{Norm of noise before inverse filter}} \leq \frac{||\mathbf{A}^{-1}||*||\mathbf{w}||}{||\mathbf{w}||}=||\mathbf{A}^{-1}||$

It seems your reasoning is correct that the norm of the inverse filter is an upper bound on the amplification.