IB Maths AI HL 3.15 Adjacency matrices and transition matrices Question Bank

Practise IB Mathematics HL 3.15 by applying adjacency matrices and transition matrices methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[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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AHL 3.15 (HL)—Adjacency matrices and transition matrices question 1