Matrices - IB Math AI Question Bank

Question 1

HL • Paper 1

Easy

Calculator Permitted

HL • Paper 1

Easy

Calculator Permitted

Let

A=\begin{pmatrix}2&-1&3\\0&4&1\end{pmatrix},\qquad B=\begin{pmatrix}1&2\\3&0\\-2&1\end{pmatrix}

A

State the orders of $AB$ and $BA$ .

[1]

B

Calculate $AB$ .

[2]

C

Calculate $BA$ and hence explain why $A$ and $B$ do not commute.

[2]

Question 2

HL • Paper 1

Easy

Calculator Permitted

HL • Paper 1

Easy

Calculator Permitted

A shop sells three models of headphones. The numbers sold during two weeks are represented by

Q=\begin{pmatrix}40&25&30\\35&32&28\end{pmatrix}

The selling prices and unit costs, in dollars, are

p=\begin{pmatrix}12\\18\\9\end{pmatrix},\qquad c=\begin{pmatrix}7\\11\\5\end{pmatrix}

A

Calculate the revenue in each week.

[2]

B

Calculate the profit in each week.

[2]

C

State which week had the greater profit and by how much.

[1]

Question 3

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

A pair of numbers is encoded using $\mathbf{y}=A\mathbf{q}$ , where

A=\begin{pmatrix}2&1\\-1&3\end{pmatrix}

A

Find $A^{-1}$ .

[2]

B

An encoded column is $\mathbf{y}=\begin{pmatrix}17\\16\end{pmatrix}$ . Decode this column.

[2]

C

Encode the column $\begin{pmatrix}8\\4\end{pmatrix}$ .

[1]

Question 4

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The matrices $A$ , $B$ , $C$ and $X$ satisfy $AX+B=C$ , where

A=\begin{pmatrix}3&1\\2&1\end{pmatrix},\quad B=\begin{pmatrix}1&-2\\0&4\end{pmatrix},\quad C=\begin{pmatrix}9&5\\6&1\end{pmatrix}

A

Find $A^{-1}$ .

[2]

B

Solve for $X$ .

[3]

Question 5

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A manufacturer combines $x$ kg of material A, $y$ kg of material B and $z$ kg of material C. The mixture has a total mass of $100$ kg. The materials provide $2$ , $3$ and $5$ nutrient units per kg respectively, and the mixture must provide $380$ nutrient units. Their costs are $4, $6 and $3 per kg respectively, and the total cost is $410.

A

Write the information as a matrix equation of the form $A\mathbf{x}=\mathbf{b}$ .

[2]

B

Use an inverse matrix to determine $x$ , $y$ and $z$ .

[3]

C

Find the percentage by mass of material C in the mixture.

[1]

Question 6

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Consider the matrix

A=\begin{pmatrix}k&2\\3&k-1\end{pmatrix}

where $k\in\mathbb{R}$ .

A

Find the values of $k$ for which $A$ is singular.

[2]

B

Find $A^{-1}$ when $k=4$ .

[2]

Question 7

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Matrices $A$ , $B$ and $X$ satisfy $XA=B$ , where

A=\begin{pmatrix}1&2\\3&5\end{pmatrix},\qquad B=\begin{pmatrix}4&1\\-2&3\end{pmatrix}

A

Find $A^{-1}$ .

[1]

B

Solve for $X$ .

[2]

C

Explain why using $X=A^{-1}B$ is not valid.

[1]

Question 8

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

At a theatre, adult tickets cost $18, student tickets cost $12 and child tickets cost $7. A total of $250$ tickets were sold for $3470. The number of adult tickets sold was $40$ more than the number of student tickets sold.

A

Represent the information as a matrix equation, using $a$ , $s$ and $c$ for the numbers of adult, student and child tickets.

[2]

B

Use an inverse matrix to determine the number of each type of ticket sold.

[2]

C

Find the percentage of the tickets sold that were child tickets.

[1]

Question 9

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Let

M=\begin{pmatrix}4&2\\1&3\end{pmatrix}

A

Find the characteristic polynomial of $M$ .

[2]

B

Hence find the eigenvalues of $M$ .

[1]

C

Find an eigenvector corresponding to each eigenvalue.

[2]

Question 10

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A two-region model is represented by

\mathbf{s}_{n+1}=M\mathbf{s}_n,\qquad M=\begin{pmatrix}0.72&0.18\\0.28&0.82\end{pmatrix}

The total population of the two regions is $5000$ .

A

Given that $\mathbf{s}_0=\begin{pmatrix}3000\\2000\end{pmatrix}$ , find $\mathbf{s}_1$ .

[2]

B

Find the invariant state with total population $5000$ .

[2]

C

State the eigenvalue associated with the invariant state and justify your answer.

[1]

Question 11

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The matrix

M=\begin{pmatrix}k&2\\1&3\end{pmatrix}

has eigenvalue $5$ .

A

Determine the value of $k$ .

[2]

B

Find an eigenvector corresponding to the eigenvalue $5$ .

[1]

C

Find the other eigenvalue of $M$ .

[1]

Question 12

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The matrix

M=\begin{pmatrix}3&1\\2&2\end{pmatrix}

has eigenvalues $4$ and $1$ . Corresponding eigenvectors are $\begin{pmatrix}1\\1\end{pmatrix}$ and $\begin{pmatrix}1\\-2\end{pmatrix}$ respectively.

A

Write down matrices $P$ and $D$ such that $M=PDP^{-1}$ .

[2]

B

Find $P^{-1}$ .

[1]

C

Hence calculate $M^5$ .

[3]

Question 13

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Each year, $85\%$ of the residents of town A remain in A and $15\%$ move to town B. Of the residents of town B, $80\%$ remain in B and $20\%$ move to town A. Initially, town A has $6000$ residents and town B has $4000$ residents. Assume the total population remains constant. Let $\mathbf{s}_n=\begin{pmatrix}A_n\\B_n\end{pmatrix}$ , where $A_n$ and $B_n$ are the populations of towns A and B, respectively, after $n$ years.

A

Write down the transition matrix $M$ such that $\mathbf{s}_{n+1}=M\mathbf{s}_n$ .

[1]

B

Calculate the population of each town after $8$ years.

[2]

C

Determine the long-term population of each town.

[3]

Question 14

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The populations, in hundreds, of two interacting species are modelled by

\mathbf{s}_{n+1}=M\mathbf{s}_n,\qquad M=\begin{pmatrix}1.1&0.2\\0.1&0.8\end{pmatrix}

Initially, $\mathbf{s}_0=\begin{pmatrix}400\\100\end{pmatrix}$ .

A

Calculate $\mathbf{s}_1$ .

[1]

B

Calculate $\mathbf{s}_6$ .

[2]

C

Find the dominant eigenvalue of $M$ .

[1]

D

Determine the long-term ratio of the first species population to the second species population, and interpret the dominant eigenvalue.

[2]

Question 15

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Let

M=\begin{pmatrix}2&1\\1&2\end{pmatrix}

The eigenvalues of $M$ are $3$ and $1$ .

A

Find corresponding eigenvectors and hence write $M=PDP^{-1}$ .

[2]

B

Hence find an expression for $M^n$ , where $n$ is a positive integer.

[2]

C

Find the least positive integer $n$ for which the element in row $1$ , column $2$ of $M^n$ is greater than $1000$ .

[2]

Question 16

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A simplified population model is defined by

\mathbf{s}_{n+1}=M\mathbf{s}_n,\qquad M=\begin{pmatrix}1&1\\1&0\end{pmatrix},\qquad \mathbf{s}_0=\begin{pmatrix}1\\0\end{pmatrix}

A

Find the eigenvalues of $M$ in exact form.

[2]

B

Find an eigenvector corresponding to each eigenvalue.

[2]

C

Using diagonalization, calculate $\mathbf{s}_{10}$ .

[2]

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A bakery produces rolls, cakes and pies. The numbers produced on Monday and Tuesday are represented by

Q=\begin{pmatrix}120&80&60\\150&70&90\end{pmatrix}

The flour and sugar required per item, in kilograms, are represented by

R=\begin{pmatrix}0.20&0.05\\0.35&0.08\\0.15&0.03\end{pmatrix}

where the first column represents flour and the second column represents sugar.

A

I.

State the order of the matrix $QR$ .

[1]

II.

Calculate $QR$ and interpret its elements.

[2]

B

Flour costs 1.40 dollars per kilogram and sugar costs 2.20 dollars per kilogram. Calculate the ingredient cost for each day.

[2]

C

On Wednesday, $400$ items are produced, using $105$ kg of flour and $24$ kg of sugar. Let $x$ , $y$ and $z$ be the numbers of rolls, cakes and pies produced. Use an inverse matrix to determine $x$ , $y$ and $z$ .

[3]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A coding system encodes a block $\mathbf q$ of three numbers using $\mathbf y=A\mathbf q$ , where

A=\begin{pmatrix}1&2&0\\0&1&1\\2&0&1\end{pmatrix}

A

I.

Calculate $\det(A)$ .

[1]

II.

Hence explain why every encoded block can be decoded uniquely.

[2]

B

I.

Find $A^{-1}$ .

[2]

II.

Decode $\mathbf y=\begin{pmatrix}13\\13\\14\end{pmatrix}$ .

[1]

C

A second coding matrix is

B=\begin{pmatrix}1&2&0\\0&1&1\\1&3&1\end{pmatrix}

Show that this matrix is unsuitable for unique decoding.

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A two-component production column $\mathbf q$ is processed first by a packaging matrix $P$ and then by a transport matrix $T$ , where

P=\begin{pmatrix}2&1\\1&3\\0&2\end{pmatrix}, \qquad T=\begin{pmatrix}1&0.5&2\\0.2&1&0.5\end{pmatrix}, \qquad \mathbf q=\begin{pmatrix}40\\30\end{pmatrix}

A process diagram showing the production column $\mathbf q$ passing first through operation $P$ and then through operation $T$, with arrows indicating the order of multiplication. The output is shown only as an unlabeled placeholder; no numerical output vector is displayed.

A

I.

Calculate $P\mathbf q$ .

[2]

II.

Hence calculate $T(P\mathbf q)$ .

[2]

B

Calculate $TP$ and verify that $(TP)\mathbf q=T(P\mathbf q)$ .

[2]

C

The final output is $\begin{pmatrix}295\\182\end{pmatrix}$ . Use an inverse matrix to recover the original production column.

[2]

Question 20

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Two digital filters act on a signal vector. Their matrices are

S=\begin{pmatrix}1&2\\0&1\end{pmatrix},\qquad T=\begin{pmatrix}2&0\\1&1\end{pmatrix}

and the initial signal is $\mathbf{v}=\begin{pmatrix}3\\4\end{pmatrix}$ .

A

I.

Find the output when filter $S$ is applied first and filter $T$ second.

[2]

II.

Find the output when the order of the filters is reversed.

[2]

B

Calculate $ST$ and $TS$ . Hence explain why the order in which the filters are applied is significant, despite matrix multiplication being associative.

[4]

Question 21

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Three sensors measure environmental quantities $x$ , $y$ and $z$ . Their readings satisfy

\begin{aligned} 2x+y-z&=7,\\ x+3y+2z&=11,\\ 3x-y+z&=3. \end{aligned}

A

(a)

I.

Write the system as a matrix equation $A\mathbf x=\mathbf b$ .

[2]

II.

Use an inverse matrix to solve the system.

[3]

B

A recalibration changes the right-hand-side vector by $\Delta\mathbf b=\begin{pmatrix}0\\1\\-1\end{pmatrix}$ .

I.

Find the corresponding change $\Delta\mathbf x$ in the solution.

[2]

II.

Hence state the recalibrated values of $x$ , $y$ and $z$ . If you did not obtain $\Delta\mathbf x$ , use $\begin{pmatrix}-0.2\\0.4\\0\end{pmatrix}$ .

[2]

Question 22

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The matrices $A$ , $B$ , $C$ and $X$ satisfy $AXB=C$ , where

A=\begin{pmatrix}1&2\\0&1\end{pmatrix},\quad B=\begin{pmatrix}2&0\\1&1\end{pmatrix},\quad C=\begin{pmatrix}15&5\\7&3\end{pmatrix}

A

I.

Find $A^{-1}$ and $B^{-1}$ .

[2]

II.

Show that $X=A^{-1}CB^{-1}$ .

[2]

B

Hence determine $X$ .

[2]

C

A student claims that $X=B^{-1}CA^{-1}$ . Explain why this rearrangement is invalid.

[2]

Question 23

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Customers subscribe to either a basic service or a premium service. Each month, $80\%$ of basic subscribers remain on basic and $75\%$ of premium subscribers remain on premium. Initially there are $540$ basic and $360$ premium subscribers. The total number of subscribers remains constant.

A two-state transition diagram labelled Basic and Premium, showing monthly movement between and within the two subscription categories without displaying the transition matrix.

A

I.

Write down the transition matrix $M$ and initial state $\mathbf s_0$ such that $\mathbf s_{n+1}=M\mathbf s_n$ .

[2]

II.

Calculate the state after one month.

[1]

B

I.

Find the eigenvalues of $M$ and an eigenvector corresponding to each.

[3]

II.

Hence find an expression for the number $B_n$ of basic subscribers after $n$ months.

[2]

C

Hence determine the long-term numbers of basic and premium subscribers.

[2]

Question 24

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A directed network has adjacency matrix

A=\begin{pmatrix} 0&1&1&0\\ 0&0&1&1\\ 1&0&0&1\\ 0&1&0&0 \end{pmatrix}

The element in row $i$ , column $j$ is $1$ when there is a direct route from location $i$ to location $j$ , and is $0$ otherwise.

	1	2	3	4
1	0	1	1	0
2	0	0	1	1
3	1	0	0	1
4	0	1	0	0

A

I.

Calculate $A^2$ .

[2]

II.

Interpret the element in row $1$ , column $4$ of $A^2$ .

[2]

B

I.

Calculate the total number of routes of exactly three stages starting at location $2$ .

[2]

II.

State why $A^3$ and $A^2A$ must be equal.

[1]

C

Explain why an entry of $2$ in a power of $A$ does not mean that there are two direct routes between the corresponding locations.

[2]

Question 25

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Four unknown calibration constants are represented by

\mathbf x=\begin{pmatrix}w\\x\\y\\z\end{pmatrix}

They satisfy $A\mathbf x=\mathbf b$ , where

A=\begin{pmatrix} 1&1&1&1\\ 2&1&0&3\\ 0&4&1&2\\ 3&0&2&1 \end{pmatrix}, eqquad \mathbf b=\begin{pmatrix}17\\28\\31\\23\end{pmatrix}

A

I.

Calculate $\det(A)$ and explain why the system has a unique solution.

[2]

II.

Use an inverse matrix to find the calibration constants.

[2]

B

After a change in conditions, the constant column becomes

\mathbf b'=\begin{pmatrix}19\\35\\31\\28\end{pmatrix}

I.

Find the new calibration column $\mathbf x'$ .

[2]

II.

Find $\Delta\mathbf x=\mathbf x'-\mathbf x$ .

[1]

C

Verify that $A\Delta\mathbf x=\mathbf b'-\mathbf b$ and explain what this demonstrates.

[2]

Question 26

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A workshop produces standard, touring and cargo bicycles. Let $x$ , $y$ and $z$ denote the daily numbers produced. The production constraints are represented by

\begin{aligned} x+y+z&=60,\\ 2x+3y+5z&=180,\\ 4x+2y+6z&=200. \end{aligned}

A

I.

Write the constraints in the form $A\mathbf{x}=\mathbf{b}$ .

[2]

II.

Use an inverse matrix to determine the daily production of each type of bicycle.

[2]

B

A revised plan leaves the total number of bicycles unchanged but increases the right-hand sides of the second and third constraints by $6$ and $4$ respectively. Determine the revised production plan and interpret the change.

[4]

Question 27

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Three-number message blocks are encoded using $\mathbf{y}=A\mathbf{q}$ , where

A=\begin{pmatrix}1&1&0\\0&1&1\\1&0&1\end{pmatrix}

A

I.

Find $\det(A)$ and hence state whether the coding can be reversed uniquely.

[2]

II.

Decode the received block $\mathbf{y}=\begin{pmatrix}15\\19\\22\end{pmatrix}$ .

[2]

B

A second coding system uses

B=\begin{pmatrix}1&1&0\\0&1&1\\1&2&1\end{pmatrix}

Show that this coding system can lose information by finding two distinct plaintext blocks that produce the same encoded block.

[4]

Question 28

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A two-sensor calibration model is $A(k)\mathbf{x}=\mathbf{b}$ , where

A(k)=\begin{pmatrix}k&1\\4&k-2\end{pmatrix}

A

I.

Find the values of $k$ for which the calibration matrix is singular.

[2]

II.

For $k=3$ and $\mathbf{b}=\begin{pmatrix}11\\15\end{pmatrix}$ , determine $\mathbf{x}$ .

[2]

B

Explain why calibration becomes unreliable when $k$ is very close to either value found in part (a)(i), even if $A(k)^{-1}$ technically exists.

[4]

Question 29

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An economy has agricultural and manufacturing sectors. The internal consumption matrix is

C=\begin{pmatrix}0.2&0.1\\0.3&0.2\end{pmatrix}

and external demand is $\mathbf{d}=\begin{pmatrix}430\\360\end{pmatrix}$ . Total output $\mathbf{x}$ satisfies $\mathbf{x}=C\mathbf{x}+\mathbf{d}$ .

A

I.

Rearrange the model into the form $A\mathbf{x}=\mathbf{d}$ and write down $A$ .

[2]

II.

Use an inverse matrix to determine the total output required from each sector.

[2]

B

Determine the internal demand $C\mathbf{x}$ and explain why total output must exceed external demand in both sectors.

[4]

Question 30

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A directed transport network has three stations. Its adjacency matrix is

A=\begin{pmatrix}0&1&1\\1&0&1\\0&1&0\end{pmatrix}

where the element in row $i$ , column $j$ is the number of direct routes from station $i$ to station $j$ .

A directed network with three labelled stations whose arrows correspond to the adjacency matrix in the stem.

A

I.

Calculate $A^2$ .

[2]

II.

Interpret the element in row $2$ , column $2$ of $A^2$ .

[2]

B

A traveller may use at most two stages, and a journey of zero stages is allowed only when the starting and finishing stations are the same. Construct a matrix whose entries count all such journeys. Hence determine the number of permitted journeys from station $1$ to station $3$ .

[4]

Question 31

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A design point is transformed using

R=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad S=\begin{pmatrix}2&0\\0&1\end{pmatrix}

Here $R$ is a rotation and $S$ is a horizontal stretch.

A coordinate-plane design showing Region A and point P(1,3), with the origin and x-axis aligned to the y=0 gridline forming the bottom edge of Region A. Tick labels must be aligned to this axis. The design indicates that a rotation and a horizontal stretch will be applied, without showing the final images.

A

I.

Find the image of $\begin{pmatrix}1\\3\end{pmatrix}$ when $R$ is applied first and $S$ second.

[2]

II.

Find the image when the stretch is applied first and the rotation second.

[2]

B

A region has area $12$ square units. Determine its area after either composite transformation and explain why the two composites have the same area scale factor but different effects on points.

[4]

Question 32

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The amplitudes of two coupled vibration modes after $n$ cycles are represented by $\mathbf{s}_n=M^n\mathbf{s}_0$ , where

M=\begin{pmatrix}5&2\\2&2\end{pmatrix},\qquad \mathbf{s}_0=\begin{pmatrix}5\\0\end{pmatrix}

A

I.

Find the eigenvalues of $M$ .

[2]

II.

Find a corresponding eigenvector for each eigenvalue.

[2]

B

Express $\mathbf{s}_n$ in terms of $n$ . Hence determine the limiting ratio of its first component to its second component.

[4]

Question 33

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the parameter-dependent system

A(k)\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}3\\4\end{pmatrix}, \qquad A(k)=\begin{pmatrix}k&1\\2&k-1\end{pmatrix}

where $k\in\mathbb R$ .

A

I.

Find the values of $k$ for which $A(k)$ is singular.

[2]

II.

Find $A(k)^{-1}$ for all other values of $k$ .

[2]

B

I.

Explain why the system has no solution when $k=2$ .

[1]

II.

Solve the system when $k=2.01$ , giving the coordinates to three significant figures.

[2]

C

Discuss why solutions close to $k=2$ may be highly sensitive to small measurement errors.

[2]

Question 34

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

M=\begin{pmatrix}5&2\\2&2\end{pmatrix}

A

I.

Find the eigenvalues of $M$ .

[2]

II.

Find an eigenvector corresponding to each eigenvalue.

[2]

B

I.

Write matrices $P$ and $D$ such that $M=PDP^{-1}$ .

[2]

II.

Hence find an expression for $M^n$ , where $n\in\mathbb Z^+$ .

[2]

C

Find the least positive integer $n$ for which the element in row $1$ , column $2$ of $M^n$ exceeds $10^5$ .

[2]

Question 35

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The populations of two interacting organisms are modelled by

\mathbf s_{n+1}=M\mathbf s_n, \qquad M=\begin{pmatrix}1.2&0.3\\0.2&0.7\end{pmatrix}, \qquad \mathbf s_0=\begin{pmatrix}500\\200\end{pmatrix}

Population values are measured in thousands.

A

I.

Find the eigenvalues of $M$ .

[2]

II.

Find a corresponding eigenvector for each eigenvalue.

[2]

B

I.

Express $\mathbf s_0$ as a linear combination of the eigenvectors.

[2]

II.

Hence find an expression for $\mathbf s_n$ . If you did not obtain the coefficients, use $a=\frac{1200}{7}$ and $b=-\frac{100}{7}$ .

[3]

C

Determine the long-term ratio of the first population to the second population and interpret the dominant eigenvalue.

[2]

Question 36

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $k\in\mathbb R$ , let

M_k=\begin{pmatrix}k&2\\1&k+1\end{pmatrix}

A

I.

Show that the eigenvalues of $M_k$ are $k+2$ and $k-1$ .

[2]

II.

Find an eigenvector corresponding to each eigenvalue.

[2]

B

I.

Write matrices $P$ and $D_k$ such that $M_k=PD_kP^{-1}$ .

[1]

II.

Hence find an expression for $M_k^n$ .

[2]

C

For $k=0.4$ , determine the long-term ratio of the two components of $M_k^n\begin{pmatrix}4\\1\end{pmatrix}$ .

[2]

Question 37

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A simplified ranking algorithm updates the scores of two webpages using

\mathbf r_{n+1}=M\mathbf r_n, \qquad M=\begin{pmatrix}0.6&0.9\\0.4&0.1\end{pmatrix}, \qquad \mathbf r_0=\begin{pmatrix}0.5\\0.5\end{pmatrix}

A

AI.

Find the eigenvalues of $M$ .

[2]

AII.

Find an eigenvector corresponding to each eigenvalue.

[2]

B

BI.

Show that

\mathbf r_0=\frac1{13}\begin{pmatrix}9\\4\end{pmatrix}-\frac5{26}\begin{pmatrix}1\\-1\end{pmatrix}

[2]

BII.

Hence find an expression for $\mathbf r_n$ .

[2]

C

Find the least value of $n$ for which each score differs from its limiting value by less than $0.001$ .

[2]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A damped displacement model is given by

\mathbf s_{n+1}=M\mathbf s_n, eqquad M=\begin{pmatrix}0&1\\-0.18&0.9\end{pmatrix}, eqquad \mathbf s_0=\begin{pmatrix}10\\7\end{pmatrix}

The first component of $\mathbf s_n$ is the displacement, in millimetres, at step $n$ .

A

I.

Find the eigenvalues of $M$ .

[2]

II.

Find an eigenvector corresponding to each eigenvalue.

[2]

B

I.

Express $\mathbf s_0$ as a linear combination of the eigenvectors.

[2]

II.

Hence find an expression for the displacement $d_n$ .

[2]

C

Determine the first step at which the magnitude of the displacement is less than $0.1$ mm, and explain why the displacement approaches zero.

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A reversible numerical filter is represented by

F=\begin{pmatrix}1&2\\0&-1\end{pmatrix}

A column $\mathbf v$ is updated repeatedly according to $\mathbf v_{n+1}=F\mathbf v_n$ .

A

I.

Show that $F^2=I$ .

[2]

II.

Hence state $F^{-1}$ .

[1]

B

I.

Find the eigenvalues of $F$ and a corresponding eigenvector for each.

[2]

II.

Diagonalize $F$ and hence obtain a formula for $F^n$ .

[2]

C

Given $\mathbf v_0=\begin{pmatrix}4\\3\end{pmatrix}$ , determine $\mathbf v_{2025}$ and explain the periodic behaviour of the sequence.

[2]

Question 40

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Matrices $A$ , $B$ and $X$ satisfy $AXB=C$ , where

A=\begin{pmatrix}1&1\\0&2\end{pmatrix},\quad B=\begin{pmatrix}2&0\\1&1\end{pmatrix},\quad C=\begin{pmatrix}11&3\\14&2\end{pmatrix}

A

I.

Find $A^{-1}$ and $B^{-1}$ .

[2]

II.

Hence solve for $X$ .

[2]

B

A student claims that $X=B^{-1}A^{-1}C$ . Explain the error and verify the correct solution by substitution.

[4]

Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The numbers of books borrowed from two sections of a library are modelled by

\mathbf{s}_{n+1}=M\mathbf{s}_n,\qquad M=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix},\qquad \mathbf{s}_0=\begin{pmatrix}800\\200\end{pmatrix}

A

I.

Show that $1$ and $0.5$ are eigenvalues of $M$ , and find a corresponding eigenvector for each.

[2]

II.

Hence show that

\mathbf{s}_n=\begin{pmatrix}600\\400\end{pmatrix}+200(0.5)^n\begin{pmatrix}1\\-1\end{pmatrix}

[2]

B

Determine the least value of $n$ for which each component is within one book of its long-term value. Interpret the long-term state.

[4]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The numbers of active users of two competing online platforms are modelled by

\mathbf{s}_{n+1}=M\mathbf{s}_n,\qquad M=\begin{pmatrix}1.2&0.4\\0.1&0.9\end{pmatrix},\qquad \mathbf{s}_0=\begin{pmatrix}90\\10\end{pmatrix}

The components are measured in thousands.

A

I.

Find the eigenvalues of $M$ .

[2]

II.

Find an expression for $\mathbf{s}_n$ .

[2]

B

Determine the first value of $n$ for which the model predicts more than one million active users in total. State one reason why predictions far beyond this point should be treated cautiously.

[4]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A conic is represented by

\begin{pmatrix}x&y\end{pmatrix} M \begin{pmatrix}x\\y\end{pmatrix}=12, \qquad M=\begin{pmatrix}5&-2\\-2&2\end{pmatrix}

Rotated ellipse in the original x-y axes.

A

I.

Find the eigenvalues of $M$ .

[2]

II.

Find unit eigenvectors and construct an orthogonal matrix $P$ that diagonalizes $M$ .

[2]

B

Using coordinates $\begin{pmatrix}u\\v\end{pmatrix}=P^T\begin{pmatrix}x\\y\end{pmatrix}$ , find the equation of the conic in $u$ and $v$ . Hence state its shape, semi-axis lengths and area.

[4]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A parameter-dependent update matrix is

M(a)=\begin{pmatrix}a&0.2\\0.2&a\end{pmatrix}

A state is updated by $\mathbf{s}_{n+1}=M(a)\mathbf{s}_n$ .

A

I.

Find the eigenvalues and corresponding eigenvector directions of $M(a)$ .

[2]

II.

Write a diagonalization of $M(a)$ .

[2]

B

Determine the set of values of $a$ for which every initial state approaches the zero vector as $n\to\infty$ . Justify your answer using the eigenvalues.

[4]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let

M(p)=\begin{pmatrix}p&1\\1&p\end{pmatrix}

where $p\in\mathbb{R}$ .

A

I.

Find the eigenvalues and corresponding eigenvectors of $M(p)$ .

[2]

II.

Hence find a formula for $M(p)^n$ , where $n$ is a positive integer.

[2]

B

For $p=0.6$ , an initial state is $\mathbf{s}_0=\begin{pmatrix}7\\3\end{pmatrix}$ . The state evolves according to $\mathbf{s}_{n+1}=M(0.6)\mathbf{s}_n$ . Define $d_n=(\mathbf{s}_n)_1-(\mathbf{s}_n)_2$ . Find an expression for $\mathbf{s}_n$ and explain the alternating behaviour of $d_n$ .

[4]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The values, in thousands of dollars, of two linked investment funds are modelled by

\mathbf{s}_{n+1}=M\mathbf{s}_n,\qquad M=\begin{pmatrix}1.05&0.02\\0.01&1.04\end{pmatrix},\qquad \mathbf{s}_0=\begin{pmatrix}320\\140\end{pmatrix}

A

I.

Find the eigenvalues and a corresponding eigenvector for each.

[2]

II.

Find an expression for $\mathbf{s}_n$ .

[2]

B

Determine the first value of $n$ for which the combined modelled value exceeds one million dollars. Hence state the long-term ratio of the value of the first fund to the second.

[4]

Question 47

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

M=\begin{pmatrix}3&2\\1&2\end{pmatrix}

A

I.

Find the characteristic polynomial and eigenvalues of $M$ .

[2]

II.

Show directly that $M^2-5M+4I=0$ .

[2]

B

I.

Deduce a recurrence relating $M^n$ , $M^{n-1}$ and $M^{n-2}$ for $n\ge2$ .

[1]

II.

Using diagonalization, find an explicit expression for $M^n$ .

[3]

C

Verify that the expression from part (b)(ii) satisfies the recurrence from part (b)(i).

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A two-page ranking model uses the transition matrix

M(d)=\begin{pmatrix}0.5&0.5+0.5d\\0.5&0.5-0.5d\end{pmatrix}

where $0<d<1$ . A ranking vector has components summing to $1$ and is updated by $\mathbf{r}_{n+1}=M(d)\mathbf{r}_n$ .

A two-page directed-link diagram illustrating a ranking process with a damping parameter, without displaying the invariant ranking.

A

I.

Show that $1$ is an eigenvalue and determine the invariant ranking vector $\mathbf{r}_*$ in terms of $d$ .

[2]

II.

Find the second eigenvalue and a corresponding eigenvector.

[2]

B

For $d=0.8$ and $\mathbf{r}_0=\begin{pmatrix}1\\0\end{pmatrix}$ , determine the least value of $n$ for which both components of $\mathbf{r}_n$ are within $0.001$ of their invariant values. Explain why the rankings alternate around equilibrium.

[4]