Projection with a system involving a singular matrix

Assume that I have a system given as $$\begin{bmatrix} A& -B\\ 0 & C \end{bmatrix} \begin{bmatrix} A^{-1}& 0\\ 0 & I\\ \end{bmatrix} \begin{bmatrix} D & B\\ 0 & I\\ \end{bmatrix} \begin{bmatrix} q_1\\ q_2\\ \end{bmatrix} = \begin{bmatrix} b_1\\ b_2\\ \end{bmatrix}$$ which I would like to solve for the $$q$$ vector. If $$A$$ is a regular matrix, there is no problem . However, my problem is related to the cases where $$A$$ can become a singular matrix. In these singular cases, I was wondering if I could develop a projection matrix, say $$T$$, which I can explicitly separate $$q_1$$ into two spaces. Namely, one component for the particular solution that would result from the solution of the singular system such as $$Ax=b$$ (when $$A$$ is singular) and the other in the null space of $$A$$, since I can not find a solution to this problem directly, the problem will always be written with the null space vector or vectors augmented with the particular solution. I am confused about this projection matrix when $$A$$ is singular.

What I would like to perform at the end is to find a transformation like

$$\begin{bmatrix} q_1\\ q_2\\ \end{bmatrix} = T \begin{bmatrix} \tilde{q}_1\\ \alpha \\ q_2\\ \end{bmatrix}$$ where $$\alpha$$ represents the coordinate related the null space and $$\tilde{q}_1$$ will be related to the coordinate space where I can use the pseudoinverse to find the particular solution.