Could you help me with this task? I should decide which of the two following statements in (i) and (ii) are true. I would appreciate it, if you would explain to me the solution process in detail or give me hints to solve the task, because I want to try to understand the task.
(i) $M^{-1}\in [\{M^i: i\geq0\}]$ for all $M\in\mathsf{GL}(7,\mathbb{C})$.
(ii) There is $M\in\mathsf{Mat}(2\times 2, \mathbb{R})$, so that $Adj(M)∉[{E2,M}]$, where $E2$ is the identity matrix.
Thanks in advance!
(i) Take the identity matrix, and replace the top-left 1 with (1/2). Can $M^i=M^{-1}$ for some $i\geq 0$?
(ii) You can just work this one out. The adjoint of $\pmatrix{a&b\\c&d}$ is $\pmatrix{d&-b\\-c&a}$. So try looking for such an $M.$