I'm reading the Introduction to Abstract Algebra by W. Keith Nicholson. In the matrix inverses section for the proving many properties, implicitly it has been used the following statement:      If a square matrix is in the reduced row echelon form, it is either   identity matrix or has a zero row.   I really have no idea about the proof and also searching wasn't useful.
I'm reading Linear Algebra by Hoffman, Kunze where the authors explained that a $n\times n$ matrix $A$ being invertible is equivalent to the fact that $A$ is row-equivalent to $n\times n$ matrix $R$ which is an identity matrix.  In the proof of the theorem, they wrote:     $$R= E_k\ldots E_2E_1 A$$ where $E_1,\ldots,E_k$ are elementary matrices. Each $E_j$ is invertible, and so $$A = E_1^{-1}\ldots E_k^{-1}~R\,. $$  ... Since, $R$ is a (square) row-reduced echelon matrix, $R$ is invertible if and only if $R=I\,.$  [...]   I couldn't get the conclusion, since any row of $R$ can't be zero, it has to be identity matrix $I\,.$ Why is it so?  Isn't there any other row-reduced echelon matrix other than the identity matrix having no zero row and invertible? Why is it so?
Assume  $$\lim_{n\to \infty}  (y_{2n} - y_n)=0$$  then  $$\lim_{n\to \infty} y_n$$  exists.  I know it's not true, and I can see a sequence that disprove that $(1,1,2,1,3,2,4,1,5,3,...)$ but I want a sequence that has formula.  hint/help?  thanks Where are the Stack Exchange data dumps? I see a lot of talk around here regarding the Stack Overflow public data dump.  Where is it? How can I play with it? What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by  $$x_1=r \cos  \theta$$ $$x_2=r \sin  \theta$$  For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by  $$x_1=r \sin  \theta  \cos  \varphi$$ $$x_2=r \sin  \theta  \sin  \varphi$$ $$x_3=r \cos  \theta$$  So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n&gt;3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks. Why is Main method private? How to reverse name and first name? With biblatex and verbose-trad2 style, I would like to reverse the order of the last name and the first name for inline references.  The default behavior is:     John DOE, Title, place : publisher, year   for inline references (in my case, in a footnote with footcite command) and:     DOE, John, Title, place : publisher, year   in the bibliography at the end of the document.  I would like to have both inline and final references in the following format (similar to the bibiography style).      DOE, John, Title, place : publisher, year   P-S : There are some close questions (like ) but only about the final bibliography (and no inline references). \DeclareNameAlias{sortname}{last-first} works only for the ending references (not inline ones). In Windows 8, when Alt+Tab, the desktop would appear as an option to switch to at the end of the list of open programs (thus Alt-Shift-Tab helpfully took you right to the desktop).  In Windows 10 that desktop option is gone. Is there any way to get it back?  I know about Windows+D but Alt+Tab is just more convenient in my opinion. And  also has no desktop and is uglier. Why do many anime not follow the manga? Why are they usually made shorter? Many anime do not follow the manga history and are made shorter. However there are a few that do follow it or try to finish the history with the manga like the big three (Naruto, Bleach, One Piece), Death Note, and FMA Brotherhood. But others finish the anime without the real ending like FMA, Ouran High School, Pandora Hearts, Tsubasa Reservoir Chronicle. Why is this so? Is it because of the high cost? Or the lack of popularity? SQL query to convert columns into rows I had a Windows 7 installation on my 1TB SSD laptop with 4 primary partitions. I wanted to install Lubuntu Linux but the process failed with an error which kind of said:     cannot write MBR   It gave me an option to put an MBR somewhere else or to boot without MBR. None of these options worked and it became a useless install. After this failed dual boot attempt, Windows would give a BSOD wit Code 007 ending - translated to inaccessible boot device. That is a huge NTFS partition and I don't think I have any other media to back it up before formatting it.  My only option is somehow rebuilding the old MBR. Hence I started creating a whole bunch of live USB boot environments  including Super GRUB Disk and MiniTool Partition Editor Live CD, but none of this could ever help me recover my lost MBR.  Win PE 4.0 reported the partition as bad disc and the latest version did not even have an option for MBR. I managed to create a live USB with Testdisk Running on Ubuntu. Whatever partition was seen as HPFS on quick  search was written in the MBR and after this the disc is totally unreadable. Earlier I could see my data through Linux and now I can see nothing.  Here are my questions:   When some data could be read (but Windows would not boot) the MBR was shown as dynamic MBR. As far as I know dynamic MBR is a Windows concept and my Linux attempt would not have converted it. While installing Linux I tried to delete 2 primary partitions which I found to be not really useful (I backed up their data first). So I am not sure what created the dynamic MBR.  Did my dual install convert those basic MBR to dynamic MBR? How did that happen? Here is a screenshot of my deep search attempt with TestDisk which is still in progress. I can see 4 NTFS partitions and one of these two system partitions should be the original MBR. When I tried to write both these system partitions (after stopping the deep search momentarily) I got an error like "partition not found". The file system utility showed me that there is no backup sector and that may be the reason why it cannot be read.  What other Resorts do I have, other than doing a plain format and then using some recovery tool to recover all my lost data? I am not even sure I will recover the folder names.  What other options should I be trying to restore my old MBR so Windows can boot?      Update  as of last I managed to create a live USB w/ W7  I used AOMEI Partition assistant 6.0 and on deeper  search I can see the 909 gb partition - it wont let me select it for recovery. I am sorry if my earlier Q was not clear. I referred to restoring an MBR from a partition . What happened earlier  - superficial search revealed all old partitions but NOT this one   and the later partitions showed up on Deep search &amp; I can select Other partitions for recovery but for this one. It wont let me select the partition I want  Here are some more screen shots    So in summary all that happened is   4 primary windows 7 partition -->  wanted to install Linux so I deleted 2 non bootable partitions after backing them up  ended up with BSOD ( inaccessible boot device ) and noticed that some windows partitions showed up as Dynamic MBR  (using Live USB Linux,gparted )  Use standard testdisk procedure to write MBR from the superficial NTFS-HDFS partition.  this is where I screwed up . Hindsight I should have really done a deep search and caught the 901gb partition to write MBR. I am not sure if I can undo this action  after this - life gets worse. I am  on a ventilator ! Now C is unreadable by windows and linux both and testdisk cannot list file ( see screenshots ) and I have been using a series of live USB's with some recovery s/w but no luck. ALL I want is to recreate that 901gb partition and the boot partition and put the MBR on the latter so I can get my windows 7 back again Taking Seats on a Plane Why am I seeing two JPG files instead of RAW + JPG with a Canon 1100D? Update query cancelled by user How can I run a DSLR from AC power? My wife has a Nikon D3100 that I've been using to film, but the battery typically won't last through long shoots.  I've gathered from searching around that I can't charge the camera with USB, but I wondered if there's ANY way to do that. It seems at least possible that an adapter could exist that would fill in for the battery and plug into the wall, but I haven't been able to find such a thing.  Is there such a product or technique? It would be nice to not buy a second battery, I'm really hoping for a "set it and forget it" solution. Determinant of block matrix with commuting blocks I know that given a $2N \times 2N$ block matrix with $N \times N$ blocks like   $$\mathbf{S} = \begin{pmatrix} A &amp; B\\  C &amp; D \end{pmatrix}$$  we can calculate $$\det(\mathbf{S})=\det(AD-BD^{-1}CD)$$ and so clearly if $D$ and $C$ commute this reduces to $\det(AD-BC)$, which is a very nice property.   My question is, for a general $nN\times nN$ matrix with $N\times N$ blocks where all of the blocks commute with each other, can we find the determinant in a similar way? That is, by first finding the determinant treating the blocks like scalars and then taking the determinant of the resulting $N\times N$ matrix.  Thanks in advance! Why can't a domain's root be a CNAME? I've just read  which essentially says that when I set up DNS for example.com, the root record can't be a CNAME, it has to be an A record.  My question is, why?   I'm sure the clever people who designed DNS didn't make arbitary restrictions for no reason, but I don't see what we gain by requring root domains to be A records. I would love to just point my example.com domain to someserver.somewebhost.example and forget about it, but I can't.  Please enlighten me, billpg. Why does the classical path give the dominant contribution in the path integral? Cannot login to chat.meta.stackexchange.com; server error when trying to access chat homepage I just recreated my MSE account after having deleted it a few years ago. Now, I'm trying to log in to Meta Stack Exchange Chat, and whenever I try to do so, it just fails silently and I'm still logged out.  Additionally, in the same browser session that I'm supposedly logged out, I get a server error when accessing the chat homepage or the chat profile page of any user. I don't get any errors when trying to access any of these pages in a private session. I also was able to sign in to Stack Exchange Chat and Stack Overflow Chat without problems, since my accounts on SO and my parent site for SE chat haven't been deleted.  I think it might be because it's trying to sign me in to my old chat profile, which was not deleted, but that is associated to my old, deleted site account, which clash and thus cause it to fail. Why is empty set an open set? I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition. The definition of an open set: a set S in n-dimensional space is called open if all its points are interior points. For any prime $p > 3$, why is $p^2-1$ always divisible by 24? Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know. Homotopy invariance for cohomology I have tried to read Hatcher's  book. He proved  Theorem 2.10: If two maps $f,g :X \to Y$ are homotopic, then they induce the same homomorphism $f_* = g_*:H_n(X) \to H_n(Y)$.  One can immediately obtain following corollaries using the dual version of prism operator described in Theorem 2.10 and some facts:  Corollary 1: If $f,g :X \to Y$ are homotopic, then $f_* = g_*:\tilde H_n(X) \to \tilde H_n(Y)$ for $n&gt;0$.  Corollary 2: If $f,g :(X,A) \to (Y,B)$ are homotopic, then $f_* = g_*:H_n(X,A) \to H_n(Y,B)$.  Corollary 3: If $f,g :X \to Y$ are homotopic, then $f^* = g^*: H^n(X) \to H^n(Y)$.  Corollary 4: If $f,g :X \to Y$ are homotopic, then $f^* = g^*:\tilde H^n(X) \to \tilde H^n(Y)$ for $n&gt;0$.  Corollary 5: If $f,g :(X,A) \to (Y,B)$ are homotopic, then $f^* = g^*:H^n(X,A) \to H^n(Y,B)$.  I wonder that how can we prove the following statements:  Corollary 6: If $f,g :X \to Y$ are homotopic, then $f_* = g_*:\tilde H_0(X) \to \tilde H_0(Y)$.  Corollary 7: If $f,g :X \to Y$ are homotopic, then $f_* = g_*:\tilde H^0(X) \to \tilde H^0(Y)$.  An attempt for Corollary 7 is  Proof of Corollary 7: By Universal Coefficient Theorem, we have $$\tilde H^0(X;G) \approx Hom(\tilde H_0(X),G).$$  Hence the result follows from the dualization of the proof Corollary 6.  Any help will be appreciated.