I define RSS as
$$RSS = (Y-X\hat{\beta})'(Y - X\hat{\beta})$$
where $\hat{\beta}$ is the solution to the ridge regression problem and is given by the closed form formula
$$\hat{\beta} = (X'X + \lambda I )^{-1}X'Y$$
I'm trying to find a closed form formula to $RSS$ as defined above, but I'm struggling to do so. Specifically, I wanted to find
$$\frac{1}{n} RSS = \frac{1}{n}(Y-X\hat{\beta})'(Y - X\hat{\beta})$$
since I want to show what $\frac{1}{n}RSS$ converges to in probability.
I've tried multiple attempts. For example, I tried to use SVD but I don't think that's the right way to go. I'm at a total loss. Can I get a hint?