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".