Correct answer: (B) XᵀX+λI_p is positive definite.; (C) β̂ is multivariate normal.
Explanation
For nonzero v, vᵀ(XᵀX+λI)v=‖Xv‖²+λ‖v‖²>0, proving B. A fixed linear transform of normal Y is normal, possibly degenerate, proving C. Writing H=XᵀX+λI gives Eβ̂=H⁻¹XᵀXβ=(I−λH⁻¹)β, generally biased, and Var β̂=H⁻¹XᵀXH⁻¹ rather than H⁻¹. **B,C**.
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