Show that, for square matrices $A$ and $B$, $A+B=AB$ implies $AB=BA$.
Matrices A+B=AB implies A commutes with B
Change paper size in mid-document My document should be letterpaper and my attachments should be a4paper. I had high hopes for the \newgeometry command from the geometry package, but the documentation says      \newgeometry is almost similar to   \geometry except that \newgeometry   disables all the options specified in   the preamble and skips the   papersize-related options: landscape,   portrait and paper size options (such   as papersize, paper=a4paper and so   forth).   Any help is appreciated! Is my ESTA valid for a second trip? Significance contradiction in linear regression: significant t-test for a coefficient vs non-significant overall F-statistic I'm fitting a multiple linear regression model between 4 categorical variables (with 4 levels each) and a numerical output. My dataset has 43 observations.  Regression gives me the following $p$-values from the $t$-test for every slope coefficient: $.15, .67, .27, .02$. Thus, the coefficient for the 4th predictor is significant at $\alpha = .05$ confidence level.  On the other hand, the regression gives me a $p$-value from an overall $F$-test of the null hypothesis that all my slope coefficients are equal to zero. For my dataset, this $p$-value is $.11$.  My question: how should I interpret these results? Which $p$-value should I use and why? Is the coefficient for the 4th variable significantly different from $0$ at the $\alpha = .05$ confidence level?  I've seen a related question, , but there was an opposite situation: high $t$-test $p$-values and low $F$-test $p$-value. Honestly, I don't quite understand why we would need an $F$-test in addition to a $t$-test to see if linear regression coefficients are significantly different from zero. grub issues message at boot - "error - can't find command hwmatch" This question  is exactly what I'm asking but was closed as non reproducible. Mine is very reproducible.  Ubuntu 18.04 freshly installed on a 64-bit UEFI machine. Default grub boot options work fine, but I wanted to add a brief countdown (my normal config).  After editing /etc/default/grub as normal and running sudo update-grub, the countdown works, but I get the error shown in the question title.  I tried the solution (copying files) in the closed question but it didn't seem to fit my configuration. Frankly I found the answer confusing.  I also tried editing /etc/grub.d/10_linux to comment out the IF-THEN structure that uses hwmatch. After that I did a sudo update-grub but I still got the error.  How can I get my countdown and not see this error?  Edit per comments -  Stock grub as installed (all non-commented-out lines shown)  GRUB_DEFAULT=0 GRUB_TIMEOUT_STYLE=hidden GRUB_TIMEOUT=0 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX=""   Edited grub to give countdown (all non-commented-out lines shown)  GRUB_DEFAULT=0 GRUB_TIMEOUT_STYLE=countdown GRUB_TIMEOUT=5 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="" My son is going on 5 and starting to learn a lot about how we're created. I've been with my husband for 10 years, but nearly 6 years ago I had no other choice but to move 1000 miles away and after trying the long distance thing it didn't work so we split up.   I ended up going through a rough patch and started dating an abusive jerk. After a year I was pregnant. He threatened me trying to force me to have an abortion and I refused. His grandmother was chief of police and he scared me into moving back to Florida.  Eventually, when I was 4 months along in my pregnancy my husband and I picked up right where we left off. He loved me so much and accepted our son as if he was his own and has done an amazing job at raising him.   He enrolled in college while I was pregnant, got us a house and went above and beyond to care for us both (as after giving birth I became very ill due to rare brain and spine conditions). I've always tried to do right by my son. Since day one, even while with my husband, I tried to keep the sperm donor involved. But he wanted no part of his son.  My husband stepped up when the biological father refused. The biological father would pretend that he cared, only to tell me he never cared about him anyway and was just trying to manipulate me into stopping child support (which I did anyway after the first year: I felt it wasn't right to take his money when my son had a dad who loved, supported and wanted him from the beginning).  This is a very long story so I'll try and make it not so much of a novel. The sperm donor has made it very clear he wants nothing ever to do with my son. My husband has been Dad since the moment my son entered this world. I've always planned to tell my son as early as possible. I think now, or in a few years, it won't be as hard on him compared to us telling him when he's a teen and who knows how he'd handle it.   I just don't know what to do. We have an extremely happy loving family. My son adores his dad. We never argue, we have a very happy non-dysfunctional marriage. I just feel we have a very special family. But I refuse to lie to my son. What if we wait and someone else maliciously tells him? Or if, when he's older and finds out, he spirals out of control?  I'm so confused, I just want to do the right thing and desperately need advice. Especially from others who have been through similar circumstances. We're planning on meeting with therapists and I guess child development specialists, if someone could also point me in the right direction for what type of doctor is best suited in this situation.   Also my son is very happy. He has a wonderful life and I feel as if he would do fine, knowing how much he loves his dad and just beginning to really grasp the concept of life. I don't think it would change anything for him, if that makes sense. I've been worrying over this for 5 years and really need some answers! Easy way to turn JavaScript array into comma-separated list? I have a one-dimensional array of strings in JavaScript that I'd like to turn into a comma-separated list. Is there a simple way in garden-variety JavaScript (or jQuery) to turn that into a comma-separated list? (I know how to iterate through the array and build the string myself by concatenation if that's the only way.) Is the current obtained using Ohm's law that which passes through the resistor or that which passes through the circuit? Shouldn't the current passing through a resistor be lesser than that which passes through a circuit? My understanding is that since Current = Charges/Time. If there exists a resistance to the flow of charges, then that must mean the charges slow down, meaning that more time is required to pass through a point. So, the current should then decrease. But, since this opposition to the flow of charges doesn't exist in the ENTIRE circuit, it should really only decrease the current in the resistor, right? Am I confusing the resistance of the resistor with the resistance of the wire? Any help will be appreciated! Thanks a bunch for lending me some of your time :D I'm still a newbie with Linux but I'm trying to learn. I have two USBs, one is 4GB and has the ISO install of Ubuntu (I think it's called live, the one with maximum of 4GB of permanent storage) and got a 64gb usb to make a permanent full install on it. I installed it following  tutorial and everything looks fine (I had to create the EFI partition with the Terminal though) but it just won't boot, the live install boots without problems.   I'm using the school's computers which have the UEFI menu password protected so I can't access that and I won't have a laptop for like 3 weeks (main reason why I'm doing this). I tested it with a friend's laptop disabling the Secure boot but it still wouldn't boot. I also saw that some tutorials don't add the EFI partition so I tried without that one and still won't work. And I know it's not the 64GB usb not booting because I had a live install on it too before and it booted fine.  Please, I have been trying to make it work for two days now, any ideas? I can barely understand this stuff so if you could explain step by step that would be awesome. Check if one list contains element from the other How to enumerate an enum How can you enumerate an enum in C#?  E.g. the following code does not compile:  public enum Suit {     Spades,     Hearts,     Clubs,     Diamonds }  public void EnumerateAllSuitsDemoMethod() {     foreach (Suit suit in Suit)     {         DoSomething(suit);     } }   And it gives the following compile-time error:     'Suit' is a 'type' but is used like a 'variable'   It fails on the Suit keyword, the second one. Is there a workaround to make CSS classes with names that start with numbers valid? I found Sean Carroll's "A No Nonsense Introduction to General Relativity" (about page .  pdf ), a 24-page overview of the topic, very helpful for beginning study.  It all got me over the hump of learning the meaning of various terms associated with GR, most of which I had heard before without understanding.  It also outlined the most important examples.  Is there a similar document for quantum field theory, which presents the main equations, briefly describes the main ideas, and summarizes the most important applications and results so that the reader can feel the lay of the land before studying in depth? mathpazo + siunitx: π turns into ß When I use mathpazo and siunitx (version 2.3h) together, if I use \pi inside the numerical argument to \SI, it will show up in the output as "ß".  \documentclass{article} \usepackage{mathpazo, siunitx} \begin{document}     $\pi + \SI{\pi}{\ohm}$ \end{document}   Produces:    But when I don't use mathpazo:    Why does this happen and how can I fix it? I'd like to generate unique random numbers between 0 and 1000 that never repeat (i.e. 6 doesn't show up twice), but that doesn't resort to something like an O(N) search of previous values to do it. Is this possible? Calculating geometry for feature class with x/y and Lat/Long values in different projections? Best books for mathematical background? Reputation and inbox notification shows up again after refresh This was asked before too. .  Currently I am facing the issue of appearance of reputation notification in above toolbar.  I can see the reputation notification, for every page refresh even after clicking and viewing the notification.  current browser: Chrome Version 36.0.1985.143 m How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$? I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete.  The lecturer then proved two things, (i) that $f(x) = \lim f_n (x)$ is bounded and (ii) that $\lim f_n \in \mathbb{R}$.  I don't understand why it's not enough that $f$ is bounded. I think the limit of a sequence of continuous functions is continuous and then if $f$ is bounded, it's in $C_b(X)$. So what is this $\lim f_n \in \mathbb{R}$ about? Many thanks for your help. Short story about a robot who wants to become a writer