Properties of the ridge estimator

For $Y \in \mathbb{R}^{n}$, $X \in \mathbb{R}^{n \times p}$, and $\beta \in \mathbb{R}^{p}$, consider a regression model $Y = X \beta + \epsilon$, where $\epsilon$ has an $n$-dimensional multivariate normal distribution with zero mean vector and identity covariance matrix. Let $I_p$ denote the identity matrix of order $p$. For $\lambda > 0$, let $\hat{\beta}_n = (X^T X + \lambda I_p)^{-1} X^T Y$, be an estimator of $\beta$. Then which of the following options is/are correct?