*I have also experimented with transforming the planes matrix to the Reduced Row Echelon Form.
*

Isn't it amazing how a simple phrase like "Reduced Row Echelon Form" can transport you back decades to when you last used it (or probably even heard/read it). Thanks for that sudden slip out of the present!

*My question is: what happens if any coefficient is zero (or actually both coefficients (e.g. a1 and a2) are zero?
*

If both are zero, then that is fine because your normal vectors are not divergent in that axis. If only one or other is zero then they are divergent and the planes are not parallel and will then intersect. You can always rotate your co-ordinate system through something other than π to prove this. That's the only one of your questions I can attempt off the top of my head, sorry.

Comment onRe: The intersection of M hyperplanes (Ndim)