IB Mathematics Applications & Interpretation Topic…

A training programme models trained and untrained staff by $s_{n+1}=M_ps_n$ , with rows and columns in that order and The policy team wants a nonzero workforce that can remain fixed, so it wants 1 to be an eigenvalue and will then check that it is dominant. The workforce is closed: there are no recruits or departures, and each employee belongs to exactly one of the two categories. Candidate matrices are to be tested for consistency with these rules; fractional headcounts represent planning expectations. The entry is intended to represent the proportion of untrained staff becoming trained in one month. The initial workforce is .

(a) the required policy value $p$ .

(b) Find both eigenvalues of the designed matrix and verify which is dominant.

(e) the first month for which this relative size is less than $0.5\%$ .

(f) $s_3$ and the Basic proportion of the total.

(g) on the suitability of this model for the account numbers in the three-year support budget the service must now commit, a horizon far beyond the one examined so far.

Consider the matrix $W= \begin{pmatrix} 4 & -2 \\ -2 & 4 \end{pmatrix}$ .

(a) Find the eigenvectors corresponding to the eigenvalue $\lambda = 6$ for the matrix $W$ .

(b) Write down the other eigenvalue of $W$ .

The day-to-day lunch choice of a student is modelled by the transition matrix $\mathbf{T}=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix}$ , where the states are Cafeteria and Food Truck in that order. The eigenvalues of $\mathbf{T}$ are $1$ and $0.5$ . If the student chooses the Cafeteria on one day, the probability of choosing the Food Truck the next day is $0.2$ . If the student chooses the Food Truck, the probability of choosing the Cafeteria the next day is $0.3$ .

A subscription service models Basic and Plus accounts by $s_{n+1}=Ms_n$ , with $s_0=\begin{pmatrix}500\\300\end{pmatrix}$ and $M=\begin{pmatrix}\tfrac{207}{250} & \tfrac{189}{500}\\\tfrac{63}{250} & \tfrac{351}{500}\end{pmatrix}.$ The analyst has established the distinct eigenpairs $\lambda_1=1.08$ , $v_1=\begin{pmatrix}3\\2\end{pmatrix}$ and $\lambda_2=0.45$ , $v_2=\begin{pmatrix}1\\-1\end{pmatrix}$ . The relative size of the second mode in the Basic component means the absolute second-mode contribution divided by the absolute dominant-mode contribution in that component.

(a) Verify both stated eigenpairs by multiplication.

(b) Find the coefficients of $s_0$ in the eigenvector basis.

(d) Show that the relative second-mode size in the Basic component is $\dfrac1{24}(\dfrac5{12})^n$ .

Consider the matrix $C = \begin{pmatrix} 7 & -3 \\ -3 & 7 \end{pmatrix}$ .

(a) Find the eigenvectors corresponding to the eigenvalue $\lambda = 10$ for the matrix $C$ .

(b) Write down the other eigenvalue of $C$ .

(d) Find the coefficients of $s_0$ in the eigenvector basis and a closed form for $s_n$ .

(e) the formal 10-month total for $p=0.35$ , $0.40$ and $0.45$ , and their long-run total behavior.

(f) how the model accounts for trained staff with how it accounts for untrained staff when the promotion proportion is $0.45$ , and what this means for describing that policy as one that grows the workforce.

(a) an eigenvector associated with the eigenvalue $1$ . Present your result as $\begin{pmatrix} a \\ b \end{pmatrix}$ , where $a,b\in\mathbb{Z}$ .

(b) Using your result from the previous part, or by another method, the stationary (long-term) probability that the student chooses the Cafeteria. Express your final answer as $\frac{p}{q}$ , where $p,q\in\mathbb{Z}^+$ .

IB Mathematics Applications & Interpretation AHL 1.15—eigenvalues and Eigenvectors Questionbank

IB Mathematics Applications & Interpretation AHL 1.15—eigenvalues and Eigenvectors Questionbank

IB Mathematics Applications & Interpretation AHL 1.15—eigenvalues and Eigenvectors Questionbank

IB Mathematics Applications & Interpretation AHL 1.15—eigenvalues and Eigenvectors Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 1.15—eigenvalues and Eigenvectors Questionbank