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4.06 Powers of matrices

Worksheet
Powers of matrices
1

Consider the matrix A=\begin{bmatrix} -6 & 1 & -3 \\ 4 & -9 & 3 \end{bmatrix}. Is it possible to find A^{2}? Explain your answer.

2

Consider the matrix A=\begin{bmatrix} -6 & -3 \\ 1 & 4 \end{bmatrix}. Is it possible to find A^{2}? Explain your answer.

3

Determine whether the following matrices can be squared:

a

\begin{bmatrix} 6 & 18 \\ 16 & 15 \end{bmatrix}

b

\begin{bmatrix} 3 \\ -7 \end{bmatrix}

4

Determine whether the following matrices can be cubed:

a

\begin{bmatrix} -6 & -2 & -4 \\ -9 & 3 & 8 \\ 10 & -8 & -7 \end{bmatrix}

b

\begin{bmatrix} -8 \\ 4 \\ -2 \end{bmatrix}

5

For each of the following matrices find A^{2}:

a

A=\begin{bmatrix} 8 & 3 \\ 7 & 5 \end{bmatrix}

b

A=\begin{bmatrix} -4 & 4 \\ 9 & -3 \end{bmatrix}

c

A=\begin{bmatrix} 3 & 6 & 4 \\ 2 & 5 & 4 \\ 6 & 5 & 2 \end{bmatrix}

d

A=\begin{bmatrix} -5 & -2 & 1 \\ 3 & 4 & 4 \\ 5 & 3 & -4 \end{bmatrix}

6

Find the missing element in the given matrices:

a

{\begin{bmatrix} 5 & 2 \\ 6 & -4 \end{bmatrix}}^2 = \begin{bmatrix} 37 & 2 \\ 6 & ⬚ \end{bmatrix}

b

{\begin{bmatrix} 3 & 6 \\ -1 & -4 \end{bmatrix}}^2 = \begin{bmatrix} 3 & -6 \\ 1 & ⬚ \end{bmatrix}

7

Given A=\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, find A^{3}.

8

Consider the matrix {\begin{bmatrix} a & b \\ c & d \end{bmatrix}}^2.

a

Determine whether the following is equal to the above matrix:

i

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}

ii

2 \begin{bmatrix} a & b \\ c & d \end{bmatrix}

iii

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} b & a \\ d & c \end{bmatrix}

b

Find the expression that would end up in the position marked X below after squaring the matrix:

{\begin{bmatrix} a & b \\ c & d \end{bmatrix}}^2 = \begin{bmatrix} X & ⬚ \\ ⬚ & ⬚ \end{bmatrix}

9

Consider the matrix {\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}}^2.

a

Determine whether the following is equal to the above matrix:

i

\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

ii

2 \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

iii

\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} c & b & a \\ f & e & d \\ i & h & g \end{bmatrix}

b

Find the expression that would end up in the position marked X below after squaring the matrix:

{\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}}^2 = \begin{bmatrix} X & ⬚ & ⬚ \\ ⬚ & ⬚ & ⬚ \\ ⬚ & ⬚ & ⬚ \end{bmatrix}

10

Determine the expression that goes in place of X in this equation: {\begin{bmatrix} x & y \\ z & p \end{bmatrix}}^2 = \begin{bmatrix} ⬚ & X \\ ⬚ & ⬚ \end{bmatrix}

11

Solve the matrix equation: {\begin{bmatrix} y & 11 \\ 4 & -2 \end{bmatrix}}^2 = \begin{bmatrix} 53 & -55 \\ -20 & 48 \end{bmatrix} for y.

12

Solve the matrix equation: {\begin{bmatrix} 2 & x \\ 7 & 3 \end{bmatrix}}^2 = \begin{bmatrix} 39 & 25 \\ 35 & 44 \end{bmatrix} for x.

13

Use your CAS calculator to find the indicated power:

a

A = \begin{bmatrix} 6 & 4 \\ -8 & 7 \end{bmatrix}, find A^{2}.

b

A = \begin{bmatrix} -5 & 3 \\ 2 & -1 \end{bmatrix}, find A^{7}.

c

A = \begin{bmatrix} 0.68 & 0.24 \\ 0.32 & 0.76 \end{bmatrix}, find A^{20}.

d

A = \begin{bmatrix} 10 & -8 & -1 \\ -5 & 1 & 2 \\ 9 & 0 & 7 \end{bmatrix}, find A^{3}.

e

A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, find A^{18}.

f

A = \begin{bmatrix} 9 & 1 & 0 & 3 \\ 0 & -2 & -1 & -4 \\ -3 & 0 & 3 & 2 \\ 5 & 0 & 2 & -7 \end{bmatrix}, find A^{4}.

g

A = \begin{bmatrix} 2 & 1 & 2 & 1 & 0 \\ 0 & 1 & 0 & 4 & 3 \\ 3 & 4 & 0 & 2 & 2 \\ 2 & 1 & 2 & 0 & 3 \\ 0 & 0 & 5 & 1 & 1 \end{bmatrix}, find A^{6}.

h

A = \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{bmatrix}, find A^{17}.

14

Use your CAS calculator to find the indicated power. Round each element to 3 decimal places.

a

A = \begin{bmatrix} 1.01 & 0 & 0.04 \\ 0 & 0.75 & 0.21 \\ 0.44 & 0.34 & 0.24 \end{bmatrix}, find A^{25}.

b

A = \begin{bmatrix} 0 & 0.4 & 0.7 & -0.1 \\ 0.9 & 0.8 & -0.7 & 0.2 \\ -0.1 & 0.5 & 0.5 & 0.1 \\ 0 & 0.2 & 0.2 & 0.6 \end{bmatrix}, find A^{15}.

c

A = \begin{bmatrix} 0 & 0 & 0 & 0.1 & 0 \\ 0.4 & 0 & 0.2 & 0.8 & 0 \\ 0 & 0.5 & 0 & 0 & 0.6 \\0.3 & 0.3 & 0.3 & 0 & 0 \\ 0.2 & 0 & 0.6 & 0 & 0.4 \end{bmatrix} , find A^{27}.

15

For each matrix A, given below:

i
Find A^{20}, rounding each element to three decimal places.
ii
Find A^{21}, rounding each element to three decimal places.
iii
What seems to be happening to the matrix A as it is raised to larger powers?
a
A = \begin{bmatrix} 0.1 & 0.4 & 0.3 \\ 0.2 & 0 & 0 \\ 0.7 & 0.6 & 0.7 \end{bmatrix}
b
A = \begin{bmatrix} 0 & 0.1 & 0 & 0.6 \\ 0.5 & 0 & 0.8 & 0 \\ 0 & 0.9 & 0 & 0.4 \\0.5 & 0 & 0.2 & 0 \end{bmatrix}
Applications
16

The map network below shows four towns and the paths connecting them:

The matrix A represents all of the single-step paths between the towns. Both the rows and columns are in the order: Kingston, Ashland, Greenville and then Dunham.

A = \begin{bmatrix} 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix}
a

Use your CAS calculator to calculate A^{4}, the matrix that represents all possible four-step paths between the towns.

b

How many four-step paths can be taken from Ashland to Dunham?

17

Five friends are participating in a puzzle event. As part of the rules of the event, each person can only send messages to two other people. The table below shows the possible communications:

AndrewBenClaraDarrelEleanor
Andrew01100
Ben00110
Clara00011
Darrel10001
Eleanor11000
a
Put this data into a 5 \times 5 matrix, A.
b
Use your CAS calculator to find A^{2}, the matrix that represents all messages that can be sent in two steps.
c
Can Darrel send a message to Clara in a maximum of two steps?
18

A particular website is to be designed so that all website content is available to a user within three clicks. The webpage adjacency matrix A (representing available links between different parts of the website) is given below.

If every webpage can be reached within three clicks, then the matrix B_3 = A + A^{2} + A^{3} will have no zero entries.

a
Find A^{2}
b
Find A^{3}
c
Find B_3 = A + A^{2} + A^{3}
d
Is every webpage able to be reached within three clicks?
A = \begin{bmatrix} 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \end{bmatrix}
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Outcomes

ACMGM016

use matrices, including matrix products and powers of matrices, to model and solve problems; for example, costing or pricing problems, squaring a matrix to determine the number of ways pairs of people in a communication network can communicate with each other via a third person

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