"you can prove a function is strictly convex if the 2nd derivative is strictly greater than 0"
That's in one dimension. A multivariate twice-differentiable function is convex iff the 2nd derivative matrix is positive semi-definite, because that corresponds to the directional derivative in any direction being non-negative. It's strictly convex iff the second derivative matrix is positive definite.
As you showed, the ridge loss function has second derivative $2\lambda I +2X^TX$, which is positive definite for any $\lambda>0$ because
- $\lambda I$ is positive definite for any $\lambda>0$
- $X^TX$ is positive semi-definite for any $X$
- the sum of a positive definite and positive semi-definite matrix is positive definite
If you aren't sure about any of these and want to check in more detail it's useful to know that $A$ is positive definite iff $b^TAb>0$ for all (non-zero) column vectors $b$. Because of this relationship, many matrix proofs of positive definiteness just come from writing the scalar proofs of positiveness in matrix notation (including non-trivial results like the Cramér-Rao lower bound for variances)