Gauge invariance of a Lagrangian

Question

How do I check whether or not the Lagrangian is a gauge invariant? A Lagrangian is $$ \mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2}m^2A_\mu A^\mu $$

Answer 1

You perform a gauge transformation, $A'_{\mu}=A_{\mu}+\partial_{\mu}\lambda$ for some function $\lambda$ and check if the Lagrangian is the same or not. The given Lagrangian changes under a gauge transformation, the $F_{\mu\nu}F^{\mu\nu}$ term doesn't change, but the second term does.