On my mid-term exam tonight, I had a problem which went something like this:
Which algebraic structure is $(M,*)$, where $M= \left \{ \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \right., \begin{bmatrix} 0 & 1\\ -1 &0 \end{bmatrix},\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix},\begin{bmatrix} 0 &-1 \\ 1 & 0 \end{bmatrix}\left. \right \} $ and * is matrix multiplication.
There must be some smart way of doing it, but I couldn't figure it out. In the end I made a table with all elements of M and relationship between them and then just read structure's properties. To me this just feels like excessive application of brute force.
When I was trying to prove that operation is closed, I tried with something like $x=\begin{bmatrix} a & b\\ c & d \end{bmatrix}, y= \begin{bmatrix} p & q\\ s& t \end{bmatrix}$ and multiply that and then prove that result is part of the set, but I run out of ideas on how to prove that result is part of the set.
I had basically same problem for inverse element and commutativity too.