If $A$ is matrix from $M_{n-1,n}$ such that sum of all elements in row is zero.  Prove  det$(AA^T)=nk^2$, for some $k\in\mathbb{Z}$.  I tried to solve it, but I’m really confused.   I know that $A[1, 1, \dots, 1]^t= 0 * [1, 1,\dots, 1]^t$  So zero is eigenvalue of $A$, so it's singular and $\det A=0$.   So $\det(AA^{T})=\det(A)\det(A^T)=\det(A)\det(A)=\det(A^2)=0$   Is this correct? Is there any other way to solve this?  Thank you in advance.
Problem: We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of elements in each row is zero. Prove that $\det(AA^T)=nk^2$, where $k\in \mathbb Z$.  What have I considered so far:   First I thought, since sum of all elements in each row is zero, zero is eigenvalue of $A$ but $A\in M_{(n-1)\times n}(\mathbb Z)$ confused me. I see that $AA^T$ will be $(n-1)$ by $(n-1)$, so I tried calculating $AA^T$ but I failed to see any connection.   I did some research and found  question but we had no mention of it, which makes me believe I am on the wrong track.  Thank you all for your help.
Can we start cleaning up [table] tag and burninate it ASAP Related questions:     (This led to founding of )   The tag  has become cluttered with all kinds of questions ranging from sql tables (for all DBM systems, HTML tag &lt;table&gt;, outputting content in any language (PHP, C#, LaTeX to name a few) as a table and several plugins of various languages and/or IDEs (like jTable, tablesorters and the kind).  The current count stands at 18,315 questions having this ambiguous tag. A few of possible suggestions for cleaning process:    +  to  |   +  to  |   +  to  |   +  to  |   + other-specific-DBM systems to whatever_DBM_it_is-table  +  to  |    Since I've no idea about other languages and their specific features regarding table, I'd ask the community for their input on this (such as , ,  and ). The other questions are either tagged with arrays or sql and similar tags, and in my opinion, just removing  from such questions would be enough! How do keyboard input and text output work? Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom representation of groups is more important than mathematicians.   Let us, for example, take $SU(2)$ for concreteness. Starting from the generic parametrization of a $3\times 3$ unitary matrix $U$ with $\det U=1$, and using the formula of generators $$J^i=-i\Big(\frac{\partial U}{\partial \theta_i}\Big)_{\{\theta_i=0\}}$$ one can find the $3\times 3$ matrix representation of the generators.  However, I'm looking for something else.    Given the Lie algebra $[J^i, J^j]=i\epsilon^{ijk}J^k$, is there a way that one can explicitly construct (not by guess or trial) the $3\times 3$ representations of $\{J^i\}$? Will the same procedure apply to solve other Lie algebras appearing in physics such as that of $SO(3,1)$ (or $SL(2,\mathbb{C})$)? Sum of $k {n \choose k}$ is $n2^{n-1}$ Proof that $\suṃ̣_{k=1}^{n}k {n \choose k}$ for $n \in \mathbb N$ is equal to $n2^{n-1}$.  As a hint I got that $k {n \choose k} = n {n-1\choose k-1} $.  I tried solving this by induction but, in the inductive step I'm not arriving to the correct result. I have a large external with many subfolders that I want to move all videos, pictures, and music to another hard-drive, what would be the best way to do this? :  Not ten minutes ago, I received this error, twice:  And also an &quot;Oops, something bad happened&quot; error.  I'm also getting question posting errors.  Can we fix this, please? Is there a VB.NET-Like operator in C#? Allow cross-site queries in data explorer I think it would be useful if we could join across databases in the data-explorer.  For example, I'd like to query up a list of questions that I had a hand in migrating away from SO, and see what percentage of them were subsequently closed or re-migrated.  I realise I could download the various data dumps and do this on my own machine, but it would be nice if it were allowed on the online version.  Is this possible/practical? Wrap text at any point if normal hyphenation not possible I have data (generated from a database, not manually typed in) that mostly fits into a table cell with fixed width, but sometimes overflows because there are long words for which no hyphenation is found.  The problem can be demonstrated with the following example:  \documentclass{article} \usepackage[utf8]{inputenc} \begin{document} \begin{tabular}{|p{1cm}|p{4cm}|} 1 &amp; Cell where normal hyphenation is possible ÄÖÜ \\ 2 &amp; CellwherenormalhyphenationisnotpossiblebutshouldwrapinsteadofoverflowÄÖÜ \end{tabular} \end{document}   My requirements are:   Text never overflows the cell (assuming some minimum width such that at least a handful characters have place) First priority measure to achieve 1. is the normal hyphenation If 2. fails, it is allowed to break the word at any character, so fill the line and break just before overflow would happen (disregarding normal hyphenation rules)   In the example, the overflow in cell 1 is prevented by measure 2., the normal hyphenation. The second cell overflows. I would like to have this cell treated by measure 3.  For the answer please consider:   Package seqsplit didn't help me The custom macro \wrap proposed in the answer to  comes near my wish, but can't handle the non-ascii characters ÄÖÜ (independent if they are input with utf8 input encoding or with \"A\"O\"U) As the text comes from a database, no manual editing of the cell content is wanted.   One of my ideas was to insert \linebreak[0] after every character, to tell LaTeX that a break can be made if no better opportunities exist. But this doesn't work like intended.  Thanks for your suggestions! Does a first generation iPod touch have the ability to download apps and use emojis? I am getting a 1st gen iPod touch for a friend and I was wondering if you can download like apps and games on it and if it has emojis? It would help to get an answer before I buy it! Texture messes up after extruding a modified plane I want to make some 3d models of pixel art. I started by adding a plane, adding a texture to it, added loop cuts to my plane, 15 loop cuts because it's a 16x16 texture, then I selected the blank faces and deleted them. As I extrude this plane, A &gt; E &gt; $0.1$, the texture messes up on the edge of the item, as shown below  Although not all faces mess up on the edge, a lot still do. I tried recalculating normals, unwrapping the object again, but I ended with the same problem, no change at all. Though I may have done it wrong, since I never did it before. I tried smart UV project and cube projection, but they just mess it up even more. Does anyone have a solution to this? Suppose that f is continuous on $(a,b)$ and $\lim_{x\to a^+}  f(x) = \lim_{x\to b^-} f(x) = \infty$. Prove that $f$ has a minimum on all of $(a,b)$.  I was thinking of applying the extreme value theorem, but I realized that it applies only to closed intervals. So I was thinking of defining a closed interval that is in $(a,b)$, and using the definition of one sided limits to go about the problem, but I don't know how to proceed. Help please? How to convert an MKV to AVI with minimal loss To convert an MKV to AVI, I do two things. The first thing I do is this:  ffmpeg -i filename.mkv -vcodec copy -acodec copy output.avi   or this:  ffmpeg -i filename.mkv -sameq -acodec copy output.avi   Either of these will convert the MKV to an AVI, but the problem is that the video does not play smoothly for some reason. That's fine though, because if I do one more thing it gets fixed:  ffmpeg -i output.avi -vcodec mpeg4 -b 4000k -acodec mp2 -ab 320k converted.avi   After I do this then the file plays without problem. I had success doing it this way for one file, but then I tried it on another file, and there is a slight, but noticeable loss in video quality. This is the output I get when doing the second step:  FFmpeg version 0.6.1, Copyright (c) 2000-2010 the FFmpeg developers   built on Dec 29 2010 18:02:10 with gcc 4.2.1 (Apple Inc. build 5664)   configuration:    libavutil     50.15. 1 / 50.15. 1   libavcodec    52.72. 2 / 52.72. 2   libavformat   52.64. 2 / 52.64. 2   libavdevice   52. 2. 0 / 52. 2. 0   libswscale     0.11. 0 /  0.11. 0  Seems stream 0 codec frame rate differs from container frame rate: 359.00 (359/1) -&gt; 29.92 (359/12) Input #0, avi, from 'output.avi':   Metadata:     ISFT            : Lavf52.64.2   Duration: 00:04:17.21, start: 0.000000, bitrate: 3074 kb/s     Stream #0.0: Video: mpeg4, yuv420p, 704x480 [PAR 229:189 DAR 5038:2835], 29.92 fps, 29.92 tbr, 29.92 tbn, 359 tbc     Stream #0.1: Audio: vorbis, 48000 Hz, stereo, s16 Output #0, avi, to 'converted.avi':   Metadata:     ISFT            : Lavf52.64.2     Stream #0.0: Video: mpeg4, yuv420p, 704x480 [PAR 229:189 DAR 5038:2835], q=2-31, 4000 kb/s, 29.92 tbn, 29.92 tbc     Stream #0.1: Audio: mp2, 48000 Hz, stereo, s16, 320 kb/s Stream mapping:   Stream #0.0 -&gt; #0.0   Stream #0.1 -&gt; #0.1   I just used arbitrarily large settings on the second step and it worked nicely before but not in this case. What settings should I use? Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$ Given two monotonically increasing continuous functions $f,g$ prove that  $$(b-a) \int^b_a f(x)g(x) dx &gt; \int^b_a f(x) dx \int^b_a g(x) dx,\; b&gt;a$$  what I have tried:  Let $$h(x) = (x-a)\int^x_a f(t)g(t) dt - \int^x_a f(t)dt\int^x_a g(t)dt $$ and trying to figure out that $h'&gt;0$ using MVT to reform $h'(x)$ as   $$h'(x) = (x-a)(-f(x)g(\xi_1) - g(x)f(\xi_2) + f(x)g(x) + f(\xi_3)g(\xi_3))$$ where $\xi_{1,2,3} \in (a,x)$ (That method does not seem to work) How to pass javascript value to controller..? I want to pass the java-script value to controller.I am using repeat on VF page.  VF page      &lt;apex:repeat value="{!listAdj}" var="Adj" id="hrs"&gt;       &lt;td&gt;&lt;div class="requiredInput"&gt;&lt;div class="requiredBlock"&gt;&lt;/div&gt;     &lt;apex:inputField id="decimal" value="{!Adj.Hours_Decimal__c}" onchange="setHidden5(this);" &gt;       &lt;/apex:inputField&gt;     &lt;/div&gt;&lt;/td&gt;             &lt;apex:inputField id="hours" value="{!Adj.Hours__c}" /&gt; &lt;apex:inputField id="decimal" value="{!Adj.Hours_Decimal__c}" onchange="setHidden5(this);"/&gt;              &lt;/apex:repeat&gt;   Javascript code  function setHidden5(element){     //split the id of the inputField by ':' and then use the 5th index in the array to exact position of the inputField in the apex:repeat.   var idArray = element.id.split(':');     var hiddenRep1 =document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":decimal");          var hiddenRep2 = document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":hours");       if(element.value.length &gt;=1){        document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":hours").disabled = true; **hiddenRep2.value=hidden1.value;**                }                             }     }   I want to pass the hiddenRep2.value to controller.How to pass that. Hidden2.value is same as hidden1.value + some addition subtraction.i dont know how to pass that value. Can someone me sort out this.  Thanks I never liked that SO/SE relied on the external gravatar service, so I was glad to . However, it turns out that one cannot use as profile picture a PNG file with transparent background (something that is supported by gravatar):    Above is a screen capture of my profile page, with a profile picture that has a transparent background. That background is turned into black, which I suppose is a bug.  The method of upload doesn't change anything (I uploaded it manually to imgur, but the same black background is observed). See the difference between the two following URLs:  http://i.stack.imgur.com/Da3ex.png?s=128&amp;g=1   and  http://i.stack.imgur.com/Da3ex.png   The problem is thus in the resizing code from imgur. By accident, I messed up a configuration file. How do I recover the default version without reinstalling or affecting other things? How to get attention to a post on a child meta site If I want to draw people's attention to a post on a child meta site, how can I do that? I want to get people  over at Area 51 to see what they think of the idea, because it requires opinions from the community. However, posts on the discussion zone of Area 51 rarely get activity, so it seems like I'm out of luck, unless something is changed. And from the sounds of , it seems like this is a problem on all child metas.  That post is similar, but while it talks about increasing activity in general, I am talking about how to raise participation to an individual post; the equivalent of a bounty  I can't use bounties because reputation is the same.   Could there be another way of drawing attention to individual posts? Cast ray to select block in voxel game How do you make HDDs and partitions mount at startup on Ubuntu?  I always keep my media and data files on separate partitions - one 2tb HDD and a 400 odd gig partition.  I have been trying to swap the default folder location for the home directory to my other drives.  It works when I change the /home/user_name/.config/user-dirs.dirs file to the directories I want, but only until I reboot the system. I am only assuming that the drives not mounting is the problem with it, but it would be helpful either way.  If any one knows another reason for why the directories change back to default each time that would be good also.