$$ \begin{aligned} \newcommand\argmin{\mathop{\rm arg~min}\limits} \boldsymbol{\beta}_{\text{ridge}} & = \argmin_{\boldsymbol{\beta} \in \mathcal{R^p}} \biggl[ ||\boldsymbol{y}-\boldsymbol{X\beta}||^2 + \lambda ||\boldsymbol{\beta}||^2 \biggr] \\ & = (\boldsymbol{X}^T\boldsymbol{X} + \lambda\boldsymbol{I_{p+1}})^{-1}\boldsymbol{X}^T\boldsymbol{y} \end{aligned} $$