# Is it true that $M^{-1}\in [\{M^i: i\geq0\}]$ for all $M\in\mathsf{GL}(7,\mathbb{C})$? [closed]

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.

## closed as off-topic by callculus, Namaste, Leucippus, Xander Henderson, JonMark PerryMay 16 '18 at 0:43

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – callculus, Namaste, Leucippus, Xander Henderson, JonMark Perry
If this question can be reworded to fit the rules in the help center, please edit the question.

• The title does not provide any information about the topic of the question. If someone reads the title he can imagine anything like:"The moon is 80.000 km far away from the earth." True or False. – callculus May 15 '18 at 19:34
• @callculus my first impression was that this is a question on mathematical logic. – Yanko May 15 '18 at 19:51
• @Yanko Or anything else. – callculus May 15 '18 at 19:55
• By the way, what were your thoughts when you tried to solve these questions? – callculus May 15 '18 at 20:15
• One question per post, please. Ask (i) including your effort to determine whether true or false. Then try (ii) on your own. If after trying a few things, jotting down definitions, you can't solve (ii), then and only then ask (ii)-including the work you've done in trying to tackle it yourself. Relevant definitions, your thoughts, why you think it might be true, why you think it might be false (or one of the two). Do not, though, post two questions with no effort, all in one post. – Namaste May 15 '18 at 20:51

(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 $M.$