Probability limit of ridge estimator mu hat

#1
I have been working on the following question for a while but I stucked at a certain point.
Screenshot 2021-03-21 163959.png
I have done part a and b and I found
= (I_T + D'D)y where I_T is the identity matrix with dimension T, lambda is tuning parameter and D is the second order difference matrix which is a tridiagonal and Toeplitz matrix of which the first row is [1, −2, 1, 0,... , 0]. However, I got stuck in part c and d.

Any help would be highly appreciated.