$$\begin{bmatrix} p(A \rightarrow A) & p(A \rightarrow B) \\ p(B \rightarrow A) & p(B \rightarrow B) \end{bmatrix}$$

Multiply the transition matrix T by itself 'n' times, then multiply by the initial state vector X: $$T^n * X$$

Multiply the inverse of the transition matrix by the current state vector: $$T^{-1} * X$$

The sum of entries in each column must equal 1, and all entries must be non-negative.

Multiply the transition matrix (T) by the current state vector (X): $$T * X$$

Repeatedly multiply the transition matrix by the state vector until the state vector converges. Alternatively, solve TX = X.

Multiply the transition matrix by itself, then multiply the result by the initial state vector: $$T^2 * X$$

Multiply the transition matrix by the current state vector: $$T * X$$

Multiply the inverse of the transition matrix by the current state vector: $$T^{-1} * X$$

$$p(A \rightarrow A)$$

Identify the states, determine the probabilities of transitioning between them, and arrange these probabilities in a matrix where columns sum to 1.

Raise the transition matrix to the power of the number of transitions, then multiply by the initial state vector.

Repeatedly multiply the transition matrix by an initial state vector until the state vector converges, or solve the equation TX = X.

Find the inverse of the transition matrix, then multiply it by the current state vector.

Check that all entries are non-negative and that the sum of entries in each column equals 1.

Define the states, determine transition probabilities, create the transition matrix, and define the initial state vector.

For a matrix egin{bmatrix} a & b \ c & d end{bmatrix}, the inverse is frac{1}{ad-bc} egin{bmatrix} d & -b \ -c & a end{bmatrix}.

Multiply rows of the first matrix by columns of the second matrix, summing the products to get each entry in the resulting matrix.

Each entry represents the probability of transitioning from the state represented by the column to the state represented by the row.

Find the steady-state vector, which represents the distribution of states that the system will eventually converge to.

A matrix representing probabilities of moving between states in a system.

A column vector showing probabilities of being in each state at a specific time.

A model that describes transitions between states, where the probability of each transition depends only on the current state.

The distribution of states in a Markov chain that doesn't change over time.

The probability of moving from state A to state B.

A matrix that, when multiplied by the original matrix, results in the identity matrix.

A square matrix with 1s on the main diagonal and 0s elsewhere.

Each column represents the probabilities of transitioning from a specific state to all other possible states.

To represent and analyze transitions between different states over time.

A transition matrix describes how probabilities change, while a state vector describes the probabilities at a specific time.