Finite Math Exam 4 Study Guide (copy)

matrix inverses, markov processes, game theory

Identity Matrix

A square matrix with 1's along the main diagonal and 0's everywhere else.

Identity Matrix Properties

For any matrix A, AI=A and IA=A. Inverses—

To find the inverse of a 2x2 square matrix, you can use the following formula:

A^-1 = (1 / (ad - bc)) * [d, -b]
                        [-c, a]

Where A is the 2x2 matrix and a, b, c, and d are the elements of the matrix.

Make sure that the determinant (ad - bc) is not zero, as a matrix with a zero determinant does not have an inverse.

Inverse

Two square matrices A and B are called inverses if AB = 1 and BA = I, where both matrices have the same dimensions.

Gauss-Jordan Inverse Method

A method to find the inverse of an nxn matrix by row reducing the augmented matrix [A|I] to reduced row echelon form. If the resulting matrix has the form [I|B], then B is the inverse of A. Otherwise, A is not invertible.

Writing System of Equations as Matrix Equation & Vice-Versa

A system of equations can be represented as a matrix equation by using the coefficient matrix, variable matrix, and the "equals to" matrix. The coefficient matrix contains the coefficients of the variables in each equation. The variable matrix contains the variables of the system. The "equals to" matrix contains the constants on the right side of each equation.

To convert a system of equations into a matrix equation, you can write it as: Coefficient Matrix * Variable Matrix = "Equals to" Matrix

Conversely, given a matrix equation, you can extract the coefficient matrix, variable matrix, and "equals to" matrix by separating the elements accordingly.

Markov Processes

A sequence of experiments where the outcomes depend on preceding experiments. The outcomes are called states, and the current outcome is the current state.

Stochastic or Transition Matrix

A matrix with all entries between 0 and 1, where each column sums to 1. It represents the probabilities of transitioning from one state to another.

Regular Stochastic Matrix

A stochastic matrix that has some power with all positive entries.

Stable Matrix

The long-term trend of a regular stochastic matrix.

Stable Distribution

A vector X that satisfies X = AX, where A is a stable matrix and the sum of entries in X is 1.

Absorbing State

A state in which all objects stay in that state in the next stage.

Absorbing Matrix

A matrix that contains at least one absorbing state, and there is a path from each state to an absorbing state.

Finding Stable State of Absorbing Matrix

The absorbing matrix can be rewritten in a form with I, S, 0, R (from top left, top right, bottom left, bottom right).

The stable matrix or long-term trend is I, S(I-R)^-1, 0, 0 (from top left, top right, bottom left, bottom right).

Fundamental Matrix

The matrix (I-R)^-1, where R is the submatrix of the absorbing matrix that corresponds to non-absorbing states. Its (ij)th entry gives the expected number of periods required to move from non-absorbing state j to non-absorbing state i. The sum of the jth column is the number of periods until an absorbing state is obtained.

Game

A situation where each player has a turn, and there are payoffs associated with each outcome.

Finding the Next State in Stochastic Matrix

In general—if S0 is the initial state, & A is the transition matrix, then Sn=A^nS0.

Two-Person Game

A game with two players, often denoted as R (row player) and C (column player).

Zero-Sum Game

A game where the payoff to one player is an equal loss to the other player (vice-versa).

Payoff Matrix

A matrix that holds information about all possible payoffs in a game. For a two-person, zero-sum game, each entry represents the amount paid to the row player, with a negative entry indicating a gain for the column player.

Strictly Determined

A game is strictly determined if the largest number of the row minimums equals the smallest number of the column maximums, and the value is M = m.

Saddle Point

The position in a game where the game value occurs.

Optimal Pure Strategies

Strategies that correspond to the row and column of the game value.

Expected Value

The expected value of a mixed strategy game is calculated by multiplying the row player's strategy vector R, the payoff matrix A, and the column player's strategy vector C, resulting in RAC.