Rescuing a Singular Fit with Ridge

A model is handed a design matrix whose second feature is simply the first one doubled — a length recorded in two different units. No intercept column was added:

$X = \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \end{pmatrix}, \qquad y = \begin{pmatrix} 2 \\ 5 \\ 7 \end{pmatrix}$

The plain normal equations cannot be solved here. Ridge regression repairs this by adding $\lambda I$ to the Gram matrix before inverting:

$w_{\text{ridge}} = \left(X^\top X + \lambda I\right)^{-1} X^\top y$

With $\lambda = 30$, what is $w_{\text{ridge}}$?