Is removing duplicate data necessary for Gaussian Process Regression (GPR)?

I will consider Non-Noisy Observations i.e. $y=f(x)$ Lets say we have the following data set of 5 training examples with one of the examples duplicated $(1,2,3,4,4)$ maps to $(2,4,6,8,8)$. Since for GPR we have to invert a Kernel Matrix and a Kernel matrix containing duplicate inputs will not be invertible we should remove duplicate training examples when doing GPR with non-noisy observation. Am I right in my reasoning ? Kindly comment

That said, numerically, the kernel matrix $K$ will occasionally become numerically singular if some points are too close together (but not necessarily identical). In this scenario, you can either identify and deal with the problem points (deletion, merging them, whatever) or you can some (small) noise: $\hat{K}=K+\epsilon I$. Usually $\epsilon=10^{-6}$ is sufficient for me, or you can perform a spectral decomposition of $K$ and then for each eigenvalue $\lambda_i$, replace it with $\hat{\lambda_i}=\max{\{\lambda_i, \epsilon\lambda_{\max}\}}$ for some small $\epsilon.$ The idea here is that you've effectively pinned the smallest eigenvalue of the matrix relative to the largest, and this may be a more "minimal" intervention into the matrix. This is an area where I'm not sure there are any good solutions.