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
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$2×2 matrix.
Do: Write the matrix power as a repeated multiplication and then perform matrix multiplication.
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$=$= |  |
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Practice questions
Question 1
Consider the matrix $A$A, defined below.
Which option correctly describes the result of the calculation $A^2$A2?
| $A$A= |
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$A^2$A2 cannot be found. It would require multiplying a $2\times3$2×3 matrix by a $2\times3$2×3 matrix, and the sizes of these matrices do not match for the purpose of matrix multiplication.
A$A^2$A2 can be found by squaring every element of $A$A. The resulting matrix will be
$A^2$A2= $36$36 $4$4 $4$4 $9$9 $4$4 $16$16 B$A^2$A2 can be found by the following process. First turn $A$A into a square matrix by removing the last column, then multiply the new matrix by itself. The result will be
$A^2$A2= $30$30 $16$16 $-24$−24 $-2$−2 C$A^2$A2 cannot be found, since you can only square $1\times1$1×1, $2\times2$2×2 and $3\times3$3×3 matrices and $A$A is not any of these sizes.
D
question 2
Find $A^2$A2.
| $A=$A= |
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$A^2=$A2= $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
question 3
Consider the following matrix.
| $2$2 | |||||
| $a$a | $b$b | ||||
| $c$c | $d$d |
Which of the following describes this calculation?
$a$a $b$b $a$a $b$b $c$c $d$d $c$c $d$d A$2$2 $a$a $b$b $c$c $d$d B$a$a $b$b $b$b $a$a $c$c $d$d $d$d $c$c CCalculate the value that would go in the position marked $X$X below after squaring the matrix.
$X$X $\editable{}$ $\editable{}$ $\editable{}$
Powers of matrices and using technology
To evaluate higher powers or powers of large matrices, technology such as CAS, graphics or online calculators can be very useful to perform the calculations accurately and avoid repetitive calculations. The instructions below look at how we can use a CAS calculator 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 |
Find $A^4$A4.
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 |
Find $A^4$A4.
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:
/mov4b.png)
But there are many more, including some that use the same edge more than once. Here are two more:
/mov5b.png)
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:
/mov6b.png)
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:
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:
/mov7b.png)
We then look for the entry in row $A$A, column $F$F, to find our answer:
/mov8b.png)
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:
/mov9b.png)
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:
/mov10b.png)
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 questions
question 4
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.
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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
How many four-step paths can be taken from Ashland to Dunham?