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|
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?