Moore-Penrose pseudoinverse of a matrix that is invertible in each column block

Question

4I am not in major in math. But currently I am working with a couple of matrices in the form like this: \begin{equation} \left[\begin{array}{rrrrrrrrrrrr} 1 & 0 & 0 & 0 & 0& 0& 0& 0& 0& 0& 0& 0\\ 0 & 1 &0.5 &0.5 & 0& 0& 0& 0& 0& 0& 0& 0\\ 0 & 0 &0.5 &0.5 & 1& 0.5& 0.5& 0& 0& 0& 0& 0\\ 0 & 0 & 0 & 0 & 0& 0.5& 0.5& 1& 0.5& 0.5& 0& 0\\ 0 & 0 & 0 & 0 & 0& 0& 0& 0& 0.5& 0.5& 1& 0\\ 0 & 0 & 0 & 0 & 0& 0& 0& 0& 0& 0& 0& 1 \end{array}\right] \end{equation} If we consider only nonzero elements, 1-3 columns is invertible, 4-6 columns is invertible, and so on. Is there any efficient way to compute Moore-Penrose pseudoinverse for this type of matrices?