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A payoff matrix is a tool used in game theory to understand the potential outcomes of a game based on the choices of two or more players. In essence, it is a table that describes the payoffs for each player given every possible combination of actions.

The payoff matrix includes rows representing the strategies of one player, and columns representing the strategies of the other player. Each cell in the matrix represents the result (or 'payoff') of both players choosing the corresponding strategies. Positive numbers often represent a gain for a player, while negative numbers indicate a loss.

For example, consider a simplified payoff matrix for two players, A and B, where A can choose either the first or second row and player B can choose either the first or second column. The result of their simultaneous choices would be the payoff found at the intersection of the chosen row and column. Understanding the payoff matrix is essential for players to deploy strategies that can lead to their best possible outcomes.

Game theory is a branch of mathematics and economics that studies strategic interactions among rational decision-makers. It aims to determine the optimal strategies for players in a game where each player's decision depends on the predicted actions of others. In game theory, games can be cooperative or non-cooperative, and they can model real-world scenarios from economics, political science, psychology, and many other fields.

In the context of the exercise we are dealing with, game theory is applied to find the expected payoff for players each deploying specific strategies. Each player aims to maximize their expected payoff while considering the possible strategies of the other player. Calculation of the expected payoff is critical to understanding which strategies might lead to the best outcomes under uncertain conditions.

Matrix multiplication is a fundamental operation in linear algebra where two matrices are multiplied to produce a third matrix. To multiply a matrix by another, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has dimensions equal to the number of rows in the first matrix by the number of columns in the second matrix.

In the step-by-step solution provided in the exercise, matrix multiplication is used to calculate the expected payoff of a game. The strategy matrices for the players are multiplied by the game's payoff matrix. The steps involve multiplying elements across the rows of the first matrix with the corresponding elements down the columns of the second matrix, then summing those products to produce the elements of the resulting matrix.

It is crucial for students to not just perform the mechanical process of matrix multiplication but to understand how the resulting figures represent the combined effect of differing strategies from the two players in a game theory context.