If you use your variables for real computations (as opposed to math typesetting) the I would consider labeling variables differently, as @Bill Simpson recommended too. The easiest:

Array[a, {3, 3}]

{{a[1, 1], a[1, 2], a[1, 3]}, {a[2, 1], a[2, 2], a[2, 3]}, {a[3, 1], a[3, 2], a[3, 3]}}

then an example of computation is:

Solve[a[1, 2]^3 - a[1, 2] + 1 == 0, a[1, 2]];
TraditionalForm[Column[%]]

Thank you so much for answering so quickly. Your answer was the one I needed!

Thanks, will look into it. Didnt know there would be such problems with subscripts.

It is meanwhile routine on this platform that questions come up on the use of, or problems (i.e. "unexpected behavior") with, Subscript. Sanders description of those as a "typographical construct" is just perfect!

Because in the beginning I had problems myself in proper understanding, I would like to give a little explanation:

If you have a variable (say a) and you make an assignment, that means that the variable gets an "ownvalue":

a = 3;
OwnValues[a]
(*  Out:  {HoldPattern[a]\[RuleDelayed]3}  *)

Whereas when you make the assignments:

there are no OwnValues at all! The point rather is that in a way you changed the properties of Subscript:

This is pretty much the same as if you write:

f[1] = 3;

There is no problem in doing this, but one would not want to call f[1] a "variable".

I am curious.

Is the seemingly often reported troublesome behavior of subscripted variables perhaps only an unexpected consequence of a design decision of convenience?

Or are there deep fundamental theoretical reasons that might be explained why it is just not possible for subscribed variables to behave like "ordinary" variables?

If it was just a decision of convenience, since giving people subscripts seems to compel them to use them and this behavior has for years caused confusion and unexpected problems, would there be any chance this design decision might be reconsidered and perhaps this problem could just go away? Would it break anything not relatively easily repaired if this were changed?

This is perhaps similar to the various "Form"s, which people seem compelled to use and are then confused when functions and operators do not then operate on the contained data exactly as if the Form were not present.

I vaguely remember something about a "principle of least surprise" from long long ago.