$$A = \begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$$  (i) Write $A$ as a product of 4 elementary matrices.  (ii) Write $A^{-1}$ as a product of 4 elementary matrices.  I have managed to solve for $A^{-1}$, which is: $$\begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}$$  But I am struggling to figure out how I would split each of these into 4 elementary matrices.  I know that if I can solve the 4 elementary matrices of one of them, I just need to invert them to get the elementary matrices of the other.  Any help?
Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help?
How do I round a number in JavaScript? While working on a project, I came across a JS-script created by a former employee that basically creates a report in the form of  Name : Value Name2 : Value2   etc.  The peoblem is that the values can sometimes be floats (with different precision), integers, or even in the form 2.20011E+17. What I want to output are pure integers. I don't know a lot of JavaScript, though.  How would I go about writing a method that takes these sometimes-floats and makes them integers? Interior of closure of open set in topological space If $U$ is an open set in a topological space, is it true that $U$ is the interior of the closure of itself?   If $U$ is open, it must be the interior of itself. But is it the interior of the closure of itself? In the closure, we include all points $x$ such that any open set containing $x$ also contains a point in $U$. In the interior of the closure, we take the union of all open sets contained in the closure. What is the difference between max, min and sup, inf? Is 20 watts of electricity dangerous? I have a few circumstances which invlove someone being shocked with 20 watts of electricity and whether it would be deadly. So here they are:   Would 1000 volts at 20 miliamperes of AC (2MHz frequency) be dangerous or deadly? Would 1000 volts at 20 miliamperes of DC be dangerous or deadly? Would 10 Kilo volts at 2 miliamperes of AC (2MHz frequency) be dangerous or deadly? and Would 10 Kilo volts at 2 miliamperes of DC be dangerous or deadly?   Thanks for everyones help. How to write this boolean expression using only NOR gates? I'm having some trouble understanding how I can convert a boolean expression to a NOR-gate only expression.   What I'm working with looks like \$T = A B' C + A' B C' + A B\$. How would I go into implementing it with NOR gates? 80's pop song with lyric "It's the story of my life if you wanna know" I'm trying to identify a song by a female 80's pop group.   Here are the lyrics I remember:     It's the story of my life if you want to know   a play where everything is meant to be   you're the story of my life and I can't let you go Over the past few weeks, there have been a number of complaints about users making suggested edits en masse whose quality was at least questionable.  Let an example of a low-quality edit be the addition of the web-frameworks tag to dozens of questions about the PHP web framework Laravel. No one is ever going to use it in a search, but Laravel is a framework, right?  (If you disagree about this being a low-quality edit, please  bear with me anyway - my point here is that the system through which such conflicts could be settled is broken.)   Examples          This is a toxic situation guaranteed to create strife because -   The new user is happy to have found a way to participate and earn some rep; they may be genuinely feeling that they are improving the site. The reviewers greenlighting most of their suggestions confirms them in their belief.   The reviewers usually see only one edit - and it's easy to see how they might go, "meh, this isn't a terribly useful edit, but hey, maybe it helps someone, and it's a new user. No point discouraging them."   The veteran users who check out the editor see dozens of what they perceive as low-quality edits, usually ignoring anything else that should be fixed in the post. To the veterans, the new user is systematically looking for excuses to make their pointless edits, creating a giant mess in order to  gain some rep - a mess they have to clean up, while there are no consequences for the editor, who gets to keep their ill-gotten points. While a bad question or answer can be downvoted - usually leading to a change of behaviour eventually - the edit system looks like it's wide open for gaming.   So tensions are high, and when there's dialog it can be angry. I'm not saying either side is completely right. I'm completely prepared to believe that many of these edits are in good faith, etc. But it's a mess because it's frustrating people. As said, there are no consequences for bad-faith mass editing; apart from commenting on some random post, there is no way to communicate with an editor. The outcome of a moderator flag is very much uncertain.  The absence of a clear set of rules means there's plenty of room for heated arguments.    Something needs to change here.  Either   We need super clear rules on what makes a good tag edit Tag-only edits should be allowed only for 2k+ users, or they should not gain 2 reputation points for &lt;2k users The possible 1000 rep gain for suggested edits needs to be rate-limited (say, 200 per month) (this is my favourite!) The possible 1000 rep gain for suggested edits needs to be lowered, say to 500 or 250 Rollbacks of suggested edits should result in annulment of the 2 points gained (this is terrible) There should be a system to vote on edits (yes, that's been suggested before) There should be a defined and officially sanctioned pathway, and ruleset, for moderator flags cast about these users    ... or something else. But something needs to give. File: \boot\bcd Status: 0xc00000e9 windows 8 I am having this error for quite a few months. When the problem occurs, my windows freezes at boot time with black screen. After few reset attempts, sometimes it shows the entitled error. after few attempts, the windows starts but everything become slow. If I again restart, it works well. But at a point windows never starts and I have to reinstall it. I have already reinstalled 3/4 times in the past 2/3 months. I checked for any HDD errors with EASEUS Partition Master Home Edition with surface test, chkdisk at boot time etc for my C drive and F drive but no errors found. Startup repair also fails to solve the problem.  Windows Boot Manager   Windows failed to start. A recent hardware or software change might be the cause.   To fix the problem:  &nbsp;&nbsp;1. Insert your Windows installation disc and restart your computer.  &nbsp;&nbsp;2. Choose your language settings, and then click "Next." &nbsp;&nbsp;3. Click "repair your computer."  If you don't have this disc, contact your system administrator or computer manufacturer for assistance.   &nbsp;&nbsp;File: \Boot\BCD  &nbsp;&nbsp;Status: 0xc00000e9  My laptop's info:  Model: Dell n5110 corei5 OS: Windows 8 PRO 64 bit Anti-virus:  Avast Home edition  Additional info: My Program Files, Program Files (x86) and Users directory is located at F drive and linked with symbolic link (mklink). Also I have changed these basic locations from regedit. I also use eclipse android intelx86 emulator, virtualbox and Genymotion emulator most often. Rounded corners in TikZ trees. I need to find my external IP address from a shell script. At the moment I use this function:   myip () {      lwp-request -o text checkip.dyndns.org | awk '{ print $NF }' }   But it depends on perl-libwww, perl-html-format, perl-html-tree installed. What other ways can I get my external IP? Android: No icon for Notification PHP is not working well on Ubuntu 13.10 and mcrypt is missing in phpmyadmin I've upgraded from UBUNTU 13.04 to 13.10 but I can not work with PHP pages or phpmyadmin . I've tried this way to install lamp on Ubuntu sudo apt-get install lamp-server^ phpmyadmin and I've done all of the configuration correctly after installation I've added this line  Include /etc/phpmyadmin/apache.conf to  /etc/apache2/apache2.conf then I restarted apache2   Now I have two problems:   In phpmyadmin on the bottom of the page is this error : The mcrypt extension is missing. Please check your PHP configuration  I've check  and mcrypt was in it , but in phpmyadmin it gives me error of missing .   The other problem is on PHP pages it seems like there is no PHP and it's all html because lots of PHP lines are printed in textbox's like : &lt;? echo $row['details']; ?&gt; Can anybody tell me what should I do ? Are black hole regarded as baryonic matter in cosmology? What is the best way to vertically center the content of a div when the height of the content is variable.  In my particular case, the height of the container div is fixed, but it would be great if there were a solution that would work in cases where the container has a variable height as well. Also, I would love a solution with no, or very little use of CSS hacks and/or non-semantic markup. Cardinality of the set of all (real) discontinuous functions This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6.     (Show that) The cardinality of the set of all discontinuous functions is $2^{2^{\aleph_0}}$. [Hint: Using exercise 2.5, show that $|\mathbb{R}^\mathbb{R}-C|=2^{2^{\aleph_0}}$ whenever $|C|\leq 2^{\aleph_0}$.]   Exercise 2.5, as referred to in the hint, provides the following result:     For $n&gt;0$, $n \cdot 2^{2^{\aleph_0}} = \aleph_0 \cdot 2^{2^{\aleph_0}} = 2^{\aleph_0} \cdot 2^{2^{\aleph_0}} = 2^{2^{\aleph_0}} \cdot 2^{2^{\aleph_0}} = (2^{2^{\aleph_0}})^n = (2^{2^{\aleph_0}})^{\aleph_0} = (2^{2^{\aleph_0}})^{2^{\aleph_0}}=2^{2^{\aleph_0}}$.   The only way I could answer this question was by making use of two theorems stated later  in the same textbook. First, it is easy to prove that $|\mathbb{R}^\mathbb{R}-C|$ is an infinite set, since if it was finite we would have \begin{equation}2^{2^{\aleph_0}}=|\mathbb{R}^\mathbb{R}|=|\mathbb{R}^\mathbb{R}-C|+|C| \leq n+2^{\aleph_0}=2^{\aleph_0}&lt; 2^{2^{\aleph_0}},\end{equation}which is a contradiction. Now I make use of a theorem requiring the axiom of choice: For every infinite set $S$ there exists a unique aleph $\aleph_\alpha$ such that $|S|=\aleph_\alpha$. So I let $|\mathbb{R}^\mathbb{R}-C|=\aleph_\alpha$ for some ordinal $\alpha$. Now assume $\aleph_\alpha &lt; 2^{2^{\aleph_0}}$. Here I make use of another theorem, which i am not sure I am allowed to do: For every $\alpha$ and $\beta$ such that $\alpha \leq \beta$, we have $\aleph_\alpha+\aleph_\beta=\aleph_\beta$. So I basically then say either $|\mathbb{R}^\mathbb{R}-C|+|C|=\aleph_\alpha$ or $|\mathbb{R}^\mathbb{R}-C|+|C|=2^{\aleph_0}$, depending on whether $\aleph_\alpha \leq 2^{\aleph_0}$ or vice-versa. The assumption I am making is that $2^{\aleph_0}=\aleph_1$, which as I understand it is basically equivalent to the continuum hypothesis, as we are saying that the least uncountable number is $\aleph_1$ (but by the previous theorem requiring the axiom of choice this seems reasonable?). So anyway assuming I can do that, we again have a contradiction since then we are saying $|\mathbb{R}^\mathbb{R}|=2^{\aleph_0}&lt;2^{2^{\aleph_0}}$ or $|\mathbb{R}^\mathbb{R}|=\aleph_\alpha &lt; 2^{2^{\aleph_0}}$.  My main problem with my proof is that I am not making use of the result given in exercise 2.5 as suggested in the hint. Can anyone help with a proof making use of this result (stated in the second block quote above). Spambots causing server crashes. How do I block them? Retrieving HTTP status code from loaded iframe with Javascript How can i make the grass bend when this cube hits it?  Here's the picture: I compile most of my documents via Makefiles which take care of the bibliography, indexes, etc. Often I'd like to compile different versions from the same TeX file changing only small things (e.g. a beamer presentation plus a version with the handout option).  Is there any standard mechanism for passing parameters to the TeX file so that I can just type make handout to get the handout version in the above example? Does the "for…in" loop in Javascript loop through the hashtables/elements in the order they are declared? Is there a browser which doesn't do it in order? The object I wish to use will be declared once and will never be modified.  Suppose I have:  var myObject = { A: "Hello", B: "World" };   And I further use them in:  for (var item in myObject) alert(item + " : " + myObject[item]);   Can I expect 'A : "Hello"' to always come before 'B : "World"' in most decent browsers?