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VCE 11 General 2023

6.06 Powers of matrices

Lesson

Raising a matrix to a power

The square matrix $A_{m\times m}$Am×m has the same number of rows as columns.

Any square matrix can be multiplied by itself and the result is a square matrix of the same dimension. So, we can write $A_{n\times n}\times A_{n\times n}=B_{n\times n}$An×n×An×n=Bn×n or, more simply, $A^2=B$A2=B.

A matrix can only be raised to a power when it is square. We often denote the square matrix of order $n$n as $A_n$An

 

Worked example

Example 1

Find $A^2$A2 for .

Think: $A^2$A2 is defined since $A$A is a square matrix and the result will also be a $2\times2$2×2 matrix.

Do: Write the matrix power as a repeated multiplication and then perform matrix multiplication.

 

$=$=
  $=$=
  $=$=

 

Practice questions

question 1

Find $A^2$A2.

$A=$A=
    $-4$4 $4$4    
    $9$9 $-3$3    
  1. $A^2=$A2=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    

 

Powers of matrices and using the calculator

Let's look at how we can use our calculators to evaluate matrix expressions involving powers.

Casio Classpad

How to use the CASIO Classpad to complete the following tasks regarding matrices

If $A$A$=$=    $2$2  $1$1    
   $1$1  $-3$3    
  1. Find $A^4$A4.

  2. Find $A+A^2+A^3$A+A2+A3.

TI Nspire

How to use the TI Nspire to complete the following tasks regarding matrices

If $A$A$=$=    $2$2  $1$1    
   $1$1  $-3$3    
  1. Find $A^4$A4.

  2. Find $A+A^2+A^3$A+A2+A3.

 

Application: counting number of routes with matrices

Matrices can be used to help us answer questions such as the following: How many different routes are there from $A$A to $F$F that use exactly $3$3 edges in the network below? We can try to find them, here are three:

But there are many more, including some that use the same edge more than once. Here are two more:

It can be just as hard to know when to stop looking as it is to find them all. Instead, we are going to use the network's adjacency matrix to answer this for us. Let’s start by writing it out:

If two vertices are connected by an edge, there is a $1$1 in their corresponding row and column. If they are not connected, there is a $0$0 instead.

The rule we are going to use is as follows:

Counting routes with matrices

The number of routes of length $n$n in a network from one vertex to another is equal to the entry in the start vertex’s row and end vertex’s column of the matrix $M^n$Mn, where $M$M is the adjacency matrix for the network.

Since we are asking about a route of length $3$3, we need to cube the matrix above - use a calculator or an online tool, don’t try to do it by hand. Here’s the result:

 

We then look for the entry in row $A$A, column $F$F, to find our answer:

 

There are $10$10. This doesn’t help us find the routes, but if we do go looking for them and find $10$10 of them, we know we can stop looking, at least!

This same idea works for directed networks as well. Consider the network below:

How many routes are there from $C$C to $B$B of length $7$7? We just need to take the $7$7th power of the adjacency matrix:

We can then read off the answer by looking in row $C$C, column $B$B - there are $43$43 different routes of length $7$7 from $C$C to $B$B in this network.

 

Practice question

question 2

The map below shows four towns and the paths connecting them. The matrix $A$A represents all of the single-step paths between the towns.

      To:  
        Kingston Ashland Greenville Dunham    

$A$A  $=$=

From: Kingston   $1$1 $1$1 $0$0 $1$1    
Ashland   $1$1 $0$0 $1$1 $0$0    
Greenville   $0$0 $1$1 $0$0 $1$1    
Dunham   $1$1 $0$0 $1$1 $0$0    
  1. Find $A^4$A4, the matrix that represents all possible four-step paths between the towns:

    $A^4$A4 $=$=     $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    

     

    Use your CAS calculator to calculate $A^4$A4

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

Outcomes

U1.AoS3.3

matrix arithmetic: the definition of addition, subtraction, multiplication by a scalar, multiplication, the power of a square matrix, and the conditions for their use

U1.AoS3.9

add and subtract matrices, multiply a matrix by a scalar or another matrix, and raise a matrix to a power

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