I am working on a problem involving basic abstract algebra/group theory and am getting confused. I am following an online course by Dr. Bob found here, and am currently on assignment two.
My difficulty lies with problem 1b where I am given a matrix $A=$ $\left( \begin{array}{ccc} 0 & -1 \\ 1 & 0 \\ \end{array} \right)$ and asked to find its order.
Now I am fairly sure that matrix multiplication is not commutative so this makes me suspect that there are either multiple answers or a convention we must adopt (which I dont think he mentioned). If I multiply on the right I get $A\cdot A = -I, A^3 = A\cdot A^2 = \left( \begin{array}{ccc} 0 & 1 \\ -1 & 0 \\ \end{array} \right)$, and $A^4 = A\cdot A^3 = I$ so $|A| =4$.
Now when I do this on the by multiplying on the left by $A$ I get the same answer, but my intuition says this is a coincidence because of the trivial chosen matrix.
Is it true in general that the order of elements in $GL(2,\mathbb{R})$ is the same regardless of which side you multiply on, or are there criterion when this property holds? Finally, since I'm guessing that this is just a special case situation, which side do I multiply on when asked to find the order of an element?
Thanks for the help!