Question 1a

Marks: 3

Consider the $2 \times 2$ matrix $A$ defined by

$A = (\begin{matrix} 0.1 & 0.4 \\ 0.9 & 0.6 \end{matrix})$

(a)

(i)

Find the characteristic polynomial of

A

.

(ii)

By solving an appropriate equation with the characteristic polynomial, find the eigenvalues

λ_{1}

and

λ_{2}

of

A

.

Key Concepts

Characteristic Polynomials
Eigenvalues & Eigenvectors

Question 1b

Marks: 4

Let $x_{1}$ and $x_{2}$ be the eigenvectors of $A$ corresponding to $λ_{1}$ and $λ_{2}$ respectively.

(b)

By solving the eigenvector equations

A x_{1} = λ_{1} x_{1}

and

A x_{2} = λ_{2} x_{2},

find eigenvectors

x_{1}

and

x_{2}

.

Key Concepts

Eigenvalues & Eigenvectors

Question 1c

Marks: 3

(c) Show that the answers to part (b) could alternatively have been found by solving the equations $(A - λ_{1} I) x_{1} = (\begin{matrix} 0 \\ 0 \end{matrix})$ and $(A - λ_{2} I) x_{2} = (\begin{matrix} 0 \\ 0 \end{matrix})$ , where $I$ is the $2 \times 2$ identity matrix.

Key Concepts

Eigenvalues & Eigenvectors

Question 2

Marks: 1

Find the eigenvalues and corresponding eigenvectors for the matrix $A$ defined as

$A = (\begin{matrix} - 1 & 4 \\ 1 & 2 \end{matrix})$

Key Concepts

Characteristic Polynomials
Eigenvalues & Eigenvectors

Question 3

Marks: 6

Consider the matrix $B$ defined as

$B = (\begin{matrix} 4 & - 6 \\ 1 & - 2 \end{matrix})$

Find the eigenvalues and corresponding eigenvectors of $B$ .

Key Concepts

Characteristic Polynomials
Eigenvalues & Eigenvectors

Question 4a

Marks: 3

Find the eigenvalues for each of the following matrices:

(a)

C = (\begin{matrix} - 2 & 13 \\ - 1 & 2 \end{matrix})

Key Concepts

Characteristic Polynomials
Eigenvalues & Eigenvectors

Question 4b

Marks: 3

(b)

D = (\begin{matrix} 6 & - 1 \\ 17 & - 2 \end{matrix})

Key Concepts

Eigenvalues & Eigenvectors

Question 5a

Marks: 3

Consider the matrix $M$ defined as

$M = (\begin{matrix} - 1 & k \\ 3 & - 1 \end{matrix})$

where $k \in ℝ$ is a constant.

The eigenvalues of $M$ are $2$ and $- 4$ .

(a)

Find the value of

k

.

Key Concepts

Characteristic Polynomials
Eigenvalues & Eigenvectors

Question 5b

Marks: 3

(b)

Find the eigenvectors of

M

that correspond to the two eigenvalues.

Key Concepts

Eigenvalues & Eigenvectors

Question 5c

Marks: 2

(c)

Hence write

M

in the form

P D P^{- 1}

, where

P

is a matrix of eigenvectors and

D

is a diagonal matrix of eigenvalues.

Key Concepts

Diagonalisation
Matrix Powers

Question 6a

Marks: 3

(a)

It is given that, for

n \times n

matrices

A

,

B

and

C

,

$A = B C B^{- 1}$

Use the properties of matrices and matrix inverses to show that

A^{2} = B C^{2} B^{- 1}

.

Key Concepts

Matrix Multiplication
Inverse Matrices

Question 6b

Marks: 3

Consider the matrix $M = (\begin{matrix} 3 & - 2 \\ p & 1 \end{matrix})$ , where $p \in ℝ$ is a constant and where it is given that $(\begin{matrix} 1 \\ 2 \end{matrix})$ is an eigenvector of $M$ .

(b)

Find the value of

p

.

Key Concepts

Eigenvalues & Eigenvectors

Question 6c

Marks: 5

(c)

Hence, by first finding the eigenvalues and the other eigenvector of

M

, write

M

in the form

M = P D P^{- 1}

for appropriate matrices

P

and

D

.

Key Concepts

Matrix Powers
Diagonalisation

Question 6d

Marks: 4

(d) (i) Use the result of part (c) to show that

$M^{n} = \frac{1}{3} (\begin{matrix} 2 (5^{n}) + {(- 1)}^{n} & - 5^{n} + {(- 1)}^{n} \\ - 2 (5^{n}) + 2 {(- 1)}^{n} & 5^{n} + 2 {(- 1)}^{n} \end{matrix})$

(ii)

Show that the expression for

M^{n}

in part (d)(i) gives the expected result when

n = 1

.

Key Concepts

Matrix Powers

Question 7a

Marks: 2

Exobiologists are studying two species of animals in a region of the distant planet Dirion. In the researchers’ models the population of Heliors (a predator species) is indicated by $h$ , while the population of Sklyveths (a competing predator species) is indicated by $s$ .

If the respective populations at a particular point in time are $h_{n}$ and $s_{n}$ , then the researchers’ data suggest that the populations one year later may be given by the following system of coupled equations:

$h_{n + 1} = 1.06 h_{n} - 0.16 s_{n}$

$s_{n + 1} = {- 0.04 h}_{n} + 0.94 s_{n}$

(a)

Represent the system of equations in the matrix form

x_{n + 1} = M x_{n}

.

Key Concepts

Introduction to Matrices
Solving Systems of Linear Equations with Matrices

Question 7b

Marks: 2

At the start of the study, there are 600 Heliors and 500 Sklyveths in the region.

(b)

Find the expected size of the respective populations after one year.

Key Concepts

Solving Systems of Linear Equations with Matrices

Question 7c

Marks: 8

(c)

By first finding the eigenvalues and corresponding eigenvectors of

M

write

M

in the form

P D P^{- 1}

, where

P

is a matrix of eigenvectors and

D

is a diagonal matrix of eigenvalues.

Key Concepts

Eigenvalues & Eigenvectors
Diagonalisation

Question 7d

Marks: 3

(d)

Hence show that the respective populations after

n

years are predicted by the model to be

h_{n} = 520 ({0.9}^{n}) + 80 ({1.1}^{n})

and

s_{n} = 520 ({0.9}^{n}) - 20 ({1.1}^{n})

.

Key Concepts

Matrix Powers

Question 7e

Marks: 4

(e)

Describe what the model predicts in the long term for the populations of the two species, and offer one criticism of the model based on this prediction.

Key Concepts

Matrix Powers
Exponential Functions & Graphs

DP IB Maths: AI HL

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