As far as I know, an equilibrium can involve a weakly dominated strategy, but cannot involve a strictly dominated strategy. Is there a general rule for when/if you can safely delete a weakly dominated strategy?

In order to better illustrate my question, I will work through an example.

![enter image description here](https://i.stack.imgur.com/DiRGu.jpg)

* The pure strategy nash equilibria are $(m,c)$ and $(d,l)$.

In order to proceed with finding mixed strategies, I am going to delete $\mathcal r$ and $\mathcal u$ as they seem to be weakly dominated.

![enter image description here](https://i.stack.imgur.com/v9BbO.jpg)

After solving a few simple linear equations:

$\left[ \pi^{\*},\rho^{\*},\sigma^{\*},\tau^{\*} \right]$=$\left[ 0,\frac{7}{17},\frac{4}{9},\frac{5}{9} \right]$

$\:$

**Question:** Is my approach valid?