In solving a least squares problem, I have a pseudo-inverse in the regular form $(A^TA)^{-1}A^T$ where $A = \left[\begin{array}{c} I \\ \sqrt{\lambda} B \end{array} \right]$. The solution claims that I should be able to reduce this to $(I + \lambda B^TB)^{-1}$, but perhaps my linear algebra is too rusty to see how this is done.
I've tried starting with the block matrix transpose identity $\left[ \begin{array}{c} M \\ N \end{array} \right]^T = \left[ \begin{array}{c c} M^T & N^T\end{array}\right]$ to reduce the $(A^TA)^{-1}$ term to $\left[ \begin{array}{c} I \\ \lambda B^T B \end{array} \right]^{-1}$, but I'm not sure if this is correct or where to proceed from here.