# finite or infinite order of matrices

• Oct 17th 2009, 12:09 PM
ChrisBickle
finite or infinite order of matrices
I was having trouble understanding a question on my hw and was wondering if anyone could clarify. The question is:

Let A, B be 2x2 matrices with

A = |0 1| and B = |0 -1| and A,B elements in GL2(R)
|-1 0| |1 -1|

Show that A and B both have finite order but AB does not.

I thought that to show order you take powers of the matrix until you get back to the identity and if only power that gets you to the identity is 0 then you have finite order. And the identity in GL2(R) is |1 0|
|0 1|
then how can you get A or B to the identity no matter what power you take A to you get either |0 1| or |0 1|
|-1 0| |1 0|

and there is no way to get the zeros to 1 except for taking the 0 power correct? and B raises the same issues..am i defining order right? or am i missing something?
• Oct 17th 2009, 12:24 PM
Matt Westwood
Quote:

Originally Posted by ChrisBickle
I was having trouble understanding a question on my hw and was wondering if anyone could clarify. The question is:

Let A, B be 2x2 matrices with

A = |0 1| and B = |0 -1| and A,B elements in GL2(R)
|-1 0| |1 -1|

Show that A and B both have finite order but AB does not.

I thought that to show order you take powers of the matrix until you get back to the identity and if only power that gets you to the identity is 0 then you have finite order. And the identity in GL2(R) is |1 0|
|0 1|
then how can you get A or B to the identity no matter what power you take A to you get either |0 1| or |0 1|
|-1 0| |1 0|

and there is no way to get the zeros to 1 except for taking the 0 power correct? and B raises the same issues..am i defining order right? or am i missing something?

You must be evaluating $A^2$ wrong, I make it:

$A^2 = \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix}$. Square that and you get back to the identity.
• Oct 17th 2009, 12:38 PM
ChrisBickle
sorry im still confused
|0 1|
|-1 0|

then A(squared) would be

|0 1|
|1 0|

and the 0's are on the wrong diagonal and with B

|0 -1|
|1 -1|

how would you get the -1 and 1 to 0 and the 0 to a 1?
• Oct 17th 2009, 12:55 PM
Plato
You have some misunderstandings.
• Oct 17th 2009, 01:04 PM
ChrisBickle
Oh ok its because Asquared is not square all entries in A it is matrix A x matrix A and for 3rd power etc u just multipy another one?
• Oct 17th 2009, 01:09 PM
Matt Westwood
Quote:

Originally Posted by ChrisBickle
Oh ok its because Asquared is not square all entries in A it is matrix A x matrix A and for 3rd power etc u just multipy another one?

Correct.
• Oct 17th 2009, 01:31 PM
tonio
Quote:

Originally Posted by ChrisBickle
I was having trouble understanding a question on my hw and was wondering if anyone could clarify. The question is:

Let A, B be 2x2 matrices with

A = |0 1| and B = |0 -1| and A,B elements in GL2(R)
|-1 0| |1 -1|

Show that A and B both have finite order but AB does not.

I thought that to show order you take powers of the matrix until you get back to the identity and if only power that gets you to the identity is 0 then you have finite order. And the identity in GL2(R) is |1 0|
|0 1|
then how can you get A or B to the identity no matter what power you take A to you get either |0 1| or |0 1|
|-1 0| |1 0|

and there is no way to get the zeros to 1 except for taking the 0 power correct? and B raises the same issues..am i defining order right? or am i missing something?

I just can't understand what you say in "...I thought that to show order you take powers of the matrix until you get back to the identity and if only power that gets you to the identity is 0 then you have finite order"..???

Anyway, multiply A by itself and get A^2 = -I ==> A^4 = I and A has order 4, and now check that B^3 = I, so ord(A) = 4 and ord(B) = 3...
Now show that AB is the matrix

(1 n)
(0 1)

and AB has infinite order.

Tonio
• Oct 17th 2009, 02:03 PM
ChrisBickle
Thanks for all your help guys I really appreciate it