I have recently discovered the Moore-Penrose psuedoinverse method, and I am currently testing the waters with it. I noticed if I have a system, say
$$a_1x_1=0$$ $$a_2x_1+a_3x_2=0$$ $$\vdots$$ $$a_5x_1+a_6x_2=0$$
we can write it in matrix form as $$AX=\hat{0}$$ where $$A=\begin{bmatrix}a_1 & 0\\ a_2 & a_3 \\ \vdots & \vdots\\ a_5& a_6 \end{bmatrix},\qquad X= [x_1\;\; x_2],\quad \hat{0}=\begin{bmatrix}0\\ \vdots\\ 0\end{bmatrix}$$
after doing all the necessary calculations it does indeed find an (left) inverse. My question is upon observation if $a_1\neq 0 $ then clearly $x_1=0$ is a solution and this would fix $x_2=0$, but the solutions obtained via the Moore-Penrose pseudoinverse method doesn't give these trivial solutions. Would the left inverses obtained via this method indeed be the correct solutions to this system?