# Finding the pure Nash equilibria for a bimatrix game

I have to find the pure Nash equilibria for the bimatrix game

\begin{align}\begin{pmatrix}11,10 & 6,9 &10,9\\ 11,6 & 6,6 & 9,6\\ 12,10 & 6,9 & 9,11\end{pmatrix}\end{align}

Let's denote the strategies for player 1 by $$A$$, $$B$$ and $$C$$ and the strategies of player 2 by $$X$$, $$Y$$ and $$Z$$. I tried different combinations of strategies, but for every strategy player 1 plays, player 2 has a strategy that it prefers. For this strategy, player 1 will play another strategy.

Can anyone help me find the pure Nash equilibria without eliminating weakly dominated strategies?

If $$\ A= \pmatrix{11&6&10\\11&6&9\\12&6&9}\$$ and $$\ B=\pmatrix{10&9&9\\6&6&6\\10&9&11}\$$ are the players' payoff matrices, then we have $$a_{22}=6 \ge a_{i2}\ \mbox{ for all } i\in\left\{1,2,3\right\}\ \mbox{, and}\\ b_{22}=6\ge b_{2j}\ \mbox{ for all } j\in\left\{1,2,3\right\}\ ,\ \ \ \ \ \$$ so the pure strategies $$\ i=2\$$ for the first player (i.e. the row selector), and $$\ j=2\$$ for the second player (column selector) constitute a Nash equilibrium for the game.