I'm given a matrix:
$$\begin{bmatrix} (4,3) & (5,1) & (6,2) \\ (2,1) & (8,4) & (3,6\\ (3,0) & (9,6) & (2,8) \end{bmatrix}$$
I'm supposed to find the pure strategy Nash equilibrium and Pareto efficiency in this game. I know how to approach when I need to find a mixed strategy using probability distributions, but I do not know how to approach this one. Any help?
Suppose the left elements from the pairs are the payoffs for player $1$, and the right the ones for player $2$, then you underline the optimal choice for player $i$ ($i=1,2$), given the choice of the other player. So for your example you would have
\begin{bmatrix} (\underline{4},\underline{3}) & (5,1) & (\underline{6},2) \\ (2,1) & (8,4) & (3,\underline{6})\\ (3,0) & (\underline{9},6) & (2,\underline{8}) \end{bmatrix}
The strategy $(1,1)$ yields a pure strategy Nash equilibrium, since the strategy $1$ is optimal for both players given the choice of the other. Can you see why this works in general for $2$ player games with a finite amount of choices?