Zero Sum Games and Weakly Dominated Strategies

Question

Can you eliminate weakly dominated strategies in a zero-sum game? If yes, why?

Doesn't the order of elimination matter.

(Normally, in ordinary games in strategic form, you can only eliminate strictly dominated strategies. And if you were to eliminate weakly dominated strategies, the order would matter)

Answer 1

Here is a similar question where the accepted answer includes a zero sum game with a Nash Equilibrium where one player plays a weakly dominated strategy: Weakly dominated Nash equilibrium in a zero-sum game

In case that gets taken down, here is the game:

$$\begin{array}{r | c | c |} & L & R \\ \hline T & (1,-1) & (1,-1) \\ \hline B & (1,-1) & (0,0) \\ \hline \end{array}$$

We have that $(T,L)$ is a Nash Equilibrium even though $L$ is weakly dominated by $R$.

Answer 2

How can this zero-sum game be solved?