IB Maths AI HL Ahl 3 15 Hl Adjacency Matrices and Transition Matrices Questions

Practise IB Mathematics HL 3.15 by applying adjacency matrices and transition matrices methods to questions from the Question Bank.

Exam points

Construct and interpret adjacency matrices and weighted adjacency tables for graphs.
Use powers of an adjacency matrix to count walks of a specified length.
Construct transition matrices for strongly connected directed or undirected graphs and interpret their entries.

[Maximum number: 1]

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.
The model is of the form

\binom{X_{n+1}}{Z_{n+1}}=\boldsymbol{M}\binom{X_n}{Z_n}

where $X_n$ is the probability of the gene being in its normal state after dividing for the nth time, and $Z_n$ is the probability of it being in another state after dividing for the nth time, where $n\in\mathbb{N}$ .
Matrix $\boldsymbol{M}$ is found to be $\left(\begin{smallmatrix}0.94&b\\0.06&0.98\end{smallmatrix}\right)$ .

Write down the value of b.

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IB Maths AI HL Ahl 3 15 Hl Adjacency Matrices and Transition Matrices Questions question 1