I have got the following task: (1) For each $n \in \mathbb{N}$, specify matrices $A, B \in \mathbb{R}^{n \times n}$ for which $A B \neq B A$ is true. (2) Determine $$ M:=\left\{A \in \mathbb{R}^{2 \times 2}: A B=B A \quad \forall B \in \mathbb{R}^{2 \times 2}\right\} $$ thus the matrices $A \in \mathbb{R}^{2 \times 2}$, which commute with all matrices $B \in \mathbb{R}^{2 \times 2}$. Could someone please help me with this or give me an approach? That would be very helpful. Thanks.
Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \  \lambda \in \mathbb R\}$$  ($I_n$ is the $n \times n$ identity matrix)  I don't know how to use $E_{ij}$ (matrix with $1$ in $(i,j)$ and $0$ elsewhere) and the elementary matrix $P_{ij}$ to prove this question. Can anyone explain it please?
I’m using tikz-cd for diagrams, and have a medium-sized diagram that I want to re-use several times.  However, when I try using a definition like:  \newcommand{\myarrow}{$\begin{tikzcd} A \ar{r} &amp; B \end{tikzcd}$   I get an error:  ERROR: Package pgfbasematrix Error: Single ampersand used with wrong catcode.   Is it possible to put a tikz-cd diagram into a defined command somehow? Problem from iOS 10.1.1 to 10.2.1.  I have no photos at all on my iPhone yet in settings it says my photo library is taking up ~200MB of space. The number of space being taken up keeps changing constantly(in the settings app on my phone). In iTunes, the storage bar at the bottom does(when in the sync menu) not show photos.           What is taking up this storage, and how can I reduce it?  Troubleshooting:   I disabled iCloud photo library and iCloud photo stream, still showing the same thing. Tried doing what @User9125 said, didn't work. I had set the date all the way back when I first got my phone, nothing showed up. Rebooting I had hundreds of photos in Whatsapp, I deleted them but I didn't get the storage back. It didn't even make a dent in the amount of storage being used by Photos so I don't think a third-party app could be taking up this storage. I don't think a system app is taking this space either, because I cleared the pictures in mail and messages. I tried using (through sideloading) in Secure Mode and Fast Mode, but it did not clear up the photos. In an ordered field we have $x^2 \ge 0$, hence the equation $-1 = x^2 + y^2$ has no solution. But what about finite fields in general? What is the solutions set $$  -1 = x^2 + y^2 $$ of this equation? Find all number $n\in \mathbb N$ such that $\varphi (n)=14$ Why would I need a firewall if my server is well configured? I know that the compiler will sometimes initialize memory with certain patterns such as 0xCD and 0xDD.  What I want to know is when and why this happens.  When  Is this specific to the compiler used?  Do malloc/new and free/delete work in the same way with regard to this?  Is it platform specific?  Will it occur on other operating systems, such as Linux or VxWorks?  Why  My understanding is this only occurs in Win32 debug configuration, and it is used to detect memory overruns and to help the compiler catch exceptions.  Can you give any practical examples as to how this initialization is useful?  I remember reading something (maybe in Code Complete 2) saying that it is good to initialize memory to a known pattern when allocating it, and certain patterns will trigger interrupts in Win32 which will result in exceptions showing in the debugger.  How portable is this? How do I give players already enchanted Tools/Armour/Weapons I'm making a map where I want players to be able to choose there kits/classes. The problem I'm having with it though is giving players already enchanted Tools/Armour/Weapons.  I'm not sure if you can use the /give [player] [item] [amount] because you'll have to use the whole {} thing and I'm not sure how to make enchanted Tools/Armour/Weapons using that.  Can anyone help? Why do we say "is it" when asking a question rather than "it is"? Short story about a man stranded on a red planet with a sentient hand as his friend I remember reading a short story 10 to 15 years ago about a man stranded alone on a red planet, where nothing would be green, and so he would miss that color. There was a sentient hand to keep him company, and it talked to him. I don't remember if the hand was a native life form or his own hallucination.  The story was not very old at that time, so it may have been written in the late 90s or 00s. if $A$ subset of $B$ then $P(B)-P(A)=P(B\setminus A)\ge 0$ Let $P$ be a probability measure on $(\Omega, \mathscr{F})$, If $a,b,\in\mathscr{F}$ and $A\subset B$, then $P(B)-P(A)=P(B\setminus A)\ge 0$.  So far I have:  $A\subset B \implies B\setminus A = B\cap A^c = \emptyset$ as $A$ and $B^c$ are disjoint. Since $A\cup (B\setminus A)=A\cup (B\cap A^c)=(A\cup B)\cap (A\cup A^c)=B\cap\Omega = B$  Then for $A,B\in \mathscr{F}\implies P(A\cup (B\setminus A))=P(A)+P(B\setminus A) = P(B)$ by countable additivity.  Rearranging, $P(B)-P(A)=P(B\setminus A) \ge 0$. I've come across quite a few answers that request the asker to, for example: "send the user an email", after which the answerer will provide them with the code or other resources they're looking for.  A few examples:          Should we take action against these questions?  I feel like these answers aren't appropriate (see my answer below).  Update:  Perhaps there are some cases where this is acceptable. I've seen a few questions regarding a particular piece of commercial software where the author/developer of the software answers, telling the asker to contact either them or their organization's in-house support.  I'm on the fence about this case. If I came across a question here describing a similar problem I was having, it would be useful to me to have the support information and know that I could contact it for assistance.  I'm interested in others' opinions on this situation in particular. Non-linearity and self-coupling of gravity I have heard that non-linearity of  has to do with the fact that gravity self-couples. What does non-linearity have to do with self-coupling? Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$ Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$? 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