Is it true that there is an interesting entry in each row of a matrix with nonzero determinant? We call an entry of an $ n × n $ matrix with nonzero determinant interesting, if by changing this entry (and only this) the determinant of the matrix can be made $0$. Is it true that there is an interesting entry in each row of a matrix with nonzero determinant? I know that each entry of every matrix with nonzero determinant is not interesting. However I could not find proof if this is a case in each row.
Can the determinant of a matrix can be made $0$? An entry of an $n \times n$ matrix with nonzero determinant is defined as interesting if by changing this entry (and only this entry) the determinant of the matrix can be made $0$.   Is it true that each entry of every matrix with nonzero determinant is interesting? Is it true that there is an interesting entry in each row of a matrix with nonzero determinant?
How do I remap multi-key sequences to individual keys? I just bought an IBM Model M keyboard. While awesome, it lacks a windows key and multimedia keys. While there are many programs out there that can map one key to another, or one key to a key sequence, are there any programs that will map a keyboard sequence to a single key?  Edit: OS = Win7 Convergence of Ratio Test implies Convergence of the Root Test What is the standard/recommended font to use in papers? I looked around but did not find that anyone has asked this before, but what are the fonts that are standard/recommended while writing academic reports/papers? How can all labels of a given type of node be restyled in Tikz?  Imagine that there are nodes of different styles, and that only the nodes of style X should have red labels. Is this possible without having to change all labels one by one of nodes of style X? example of not surjective isometry on a bounded not closed set. I woke up this morning to find that I suffered a massive drop in reputation points across the network because lots of users had been removed.  What happened to those users? Was there a database cleanup? How do you declare an interface in C++? How do I setup a class that represents an interface?  Is this just an abstract base class? Some questions and answers are marked Community Wiki and are owned by a community wiki user:  (In fact, this very question is one of them)  Why have Community Wiki posts? How do Community Wiki posts work? How does a post become a Community Wiki post? How can the Community Wiki status be removed from a post? How can I find Community Wiki posts? In transformers, the ratio of the voltages equals the ratio of the turns - so double the output coil's turns and the output voltage doubles. Then, in order to conserve energy, current halves.  This makes perfect sense in terms of $\mathrm{P=VI}$, but what happened to $\mathrm{V=IR}$? Doubling voltage and halving the current seems to completely contradict this basic law. That is, of course, unless the resistance in the output circuit changes, with R proportional to $\mathrm{V^2}$ - but I don't see how this is possible.  So how can a transformer obey both laws? Can resistance change or am I missing something else? What is the definition of a symmetric distribution? What's the definition of a symmetric distribution? Someone told me that a random variable $X$ came from a symmetric distribution if and only if $X$ and $-X$ has the same distribution. But I think this definition is partly true. Because I can present a counterexample $X\sim N(\mu,\sigma^{2})$ and $\mu\neq0$.  Obviously, it has a symmetric distribution, but $X$ and $-X$ have different distribution! Am I right? Do you guys ever think about this question? What's the exact definition of symmetric distribution? I'm taking a look at  (now removed), which all seem to be pretty much the same.  Should any be downvoted/flagged, and if so, which ones? How do I turn on SharePoint App store in SharePoint 2013 Async Await Handler Deadlock Condition on a,b and c satisfying an equation(TIFR GS 2017) Let $a,b,c$ be positive real numbers satisfying $$(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=16,$$ then $a+b+c=3$.  I thought about the application of the AM-GM-HM inequality, but in vain. I also thought about splitting 16 into factors and comparing but went nowhere. Any ideas. Thanks beforehand. "Typos" are acknowledged as "technical" errors, ones that obviously aren't caused by the writer's poor spelling. Although less frequent than in typing, they can also occur in handwritten texts, as a result of absent-mindedness or just an accidental stroke of the pen.   Since the text wasn't actually typed, it sounds funny to call them "typos". Is there a more fitting word, one that reflects the "accidental" nature of the error? How can I open a text file with TextWrangler from the Terminal (command line)? How should ambiguous tags be dealt with? I have a directory which contains numbered image files, something like this:  01.png 02.png 03.png 03.svg 04.png 05.png 06.jpg 07.png 08.png 09.png 09.svg 10.png   Sometimes there may be multiple versions of a file in different formats (eg. a png and svg version of the 03 and 09 files above) but the numbers are otherwise consecutive.  Typically there are 40-80 such files in each directory.  The numbers correspond to the order these images appear in a manuscript (a Word document, but that's not important).  There is no other way to determine the order of the images.  If I add a new image to the manuscript I need to place a copy of the image in this directory with the correct numbering.  So if the new image is the fifth in the manuscript I need to rename the files in the directory to this in order to make room for it:  01.png 02.png 03.png 03.svg 04.png 06.png 07.jpg 08.png 09.png 10.png 10.svg 11.png   What is the most straightforward way from the command line, or from a script or macro to renumber all the files starting at a certain number?  I have a standard Fedora Linux install using bash. Do downvotes on Meta truly mean something different? It is often said that voting on Meta means something different than voting on regular sites. Upvotes mean "I agree", and downvotes mean "I disagree".  However, I submit that this is not the case. Downvoting of a question has one effect in particular: censorship.  Once a question reaches -8 votes, it disappears from the front page. It can still be found; it isn't deleted. But all it takes is a net 8 downvotes for a question to be more or less lost without a direct link.  You might say that a net 8 downvotes obviously means that the community doesn't agree or want that. Well... does it? Particularly on polarizing issues, a question can temporarily get -8 votes as different people come online and see it. Maybe certain people with a certain belief all see it at once and downvote it a lot.  The front page is how most people find stuff they want to talk about, vote on, or answer. Removal from the front page means that most people will not find it without being pointed directly to it. Once a question hits -8, it is unlikely to be upvoted again. Edits will be irrelevant, new answers won't matter. The question is basically exiled.  The problem is that, on polarized topics, fast mass downvoting can quash any idea that is not popular with the most active members. After all, the most active members are the ones who are most likely to be online and see it. If a bunch of MSO regulars see it and downvote it, it goes away. Someone who checks MSO maybe once a day is less likely to be able to catch the question before it is exiled. This gives undue power to members who are most active on MSO, simply because they have more free time to spend on MSO.  I think this is a significant problem for polarized topics. It becomes very difficult to adequately discuss something that a significant number of active members dislike, even if the community at large doesn't mind.  Consider . It currently stands at +9/-8. Initially, in the first few hours, the question was heavily downvoted and even closed. However, after a day, it was upvoted and reopened.  This shows that a question can initially have a negative response, but later get a positive one. And you can check; any edits I did to the question were relatively minor. The question posted as is was initially negative, but ultimately positive.   If that -8 happened to all come first, odds are good that the +9 and the reopen would never have happened. Those 9 people would never have had the chance to weigh in on the subject. And that's wrong.  It is wrong for someone's potential vote to not matter just because they don't visit MSO multiple times in a day. It is wrong for someone's potential vote to not matter just because they have a different sleep schedule from many of MSO's members.  The problem is one of timing. Heavy dislike from active members who quickly see the question and downvote it can kill a question before someone who might upvote it ever even sees it.  I suppose the best suggestion would be to either lower the -8 threshold (or remove it altogether) or keep it from kicking in until 1-2 days has passed. If the question isn't active, it'll already be off the front page. And if it is active, then enough people will have seen it to know if the threshold represents a real consensus or just the fast-acting few. How are digital movies sent to a movie theater? For many years, movies were stored on rolls of film that were physically mailed to theaters. Now, most theaters project their movies digitally. But how are these digital movies sent to the theater?   Do they mail a hard drive to the theater? Do they give the theater a username and password to a super secret website where they can download it? How does it get there?