I am trying to understand the process for solving group theory questions.
Let $a=\begin{bmatrix} 1&1\\0&1 \end{bmatrix}$ and $b=\begin{bmatrix}i&0\\0&-i\end{bmatrix}$ - 2 x 2 matrices with complex entries.
- Describe the smallest group of 2 x 2 complex matrices containing b.
- Describe the smallest group of 2 x 2 complex matrices containing a.
My answer:
Say for question 1 - To find the smallest group I should take $b$ and multiply it by itself $n$ times until I get the identity matrix. And then the identity matrix and the $b, b^2,..., b^n$ other matrices will be the smallest group of 2x2 matrices possible? Is that the correct way to do it?
What about question 1...multiplying $a$ by itself multiple times is not going to bring me back to the identity? So what is the procedure required for that question?