For the matrix B above, I'm not able to understand how they have extended to solution (1,0) to (0,1,0). I understand why this extension is necessary ( because A is a 2 by 3 matrix and so y must have 3 entries as well) but don't understand why the 0 has been added at the top, could it have been added in the second or third positions as well instead of the first so e.g we would have got (1,0,0) if 0 had been added in the second or third positions?
Similarly for this question I'm not able to understand how (2/3,1/3) has been extended to (0,2/3,1/3) and why (1/3, 2/3) has been extended to (1/3,2/3,0) . why have the zeroes been added add the front for the first extension and at the end for the second extension. I understand why these extensions are required( because A is a 3 by 3 matrix so solutions x and y need to have 3 entries each) but not able to understand where exactly we are supposed to add the zero in the extended solutions.
Any help would be much appreciated.
Removing a dominated strategy does not mean that the strategy goes away. It just means that it would be very irrational for player $i$ to play the dominated strategy because if you're Player I, then you would receive a lower payoff, and if you're Player II, you would get hurt more. (Since in zero-sum games, Player I's gain is Player II's loss.)
So you solve the game with the dominated strategies removed, but the final solution must consider the dominated strategies, but you just give it probability $0$. So for example, in your first game, you found that the first column was dominated by some mixed domination between column $2$ and $3$. You found that the solution to the resulting game with dominated strategies removed was the pure strategy, $(1,0)$. Let's interpret what this means. This means that you should play the second strategy 100% of the time, and never play the other two strategies. Therefore, strategies $1$ and $3$, corresponding to the columns of the matrix, should have probability zero. The reason $0$ must go to the "top" of the resulting strategy vector is because that's the entry corresponding to the dominated strategy. So if you wanted to, and switch the matrix to the one shown below \begin{align} \begin{bmatrix} 4 & 11 & 2 \\ 4 & 2 & 7 \end{bmatrix} \end{align} then the resulting strategy vector is $(1,0,0)$. The reason the zero must be added to the top is because the dominated strategy was the first column, hence a zero should be added to the first entry of the strategy vector. In the matrix above, the dominated strategy was in the third column, and therefore the zero should be placed in the third entry of the strategy vector.