Let $A$ and $B$ be square matrices of order $n$. Show that $AB - BA$ can never be equal to unit matrix.   How to approach above question. Please help.
The following question is from Artin's Algebra.      If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.    I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work.
How do you do load testing and capacity planning for web sites? This is a  about capacity planning for web sites. Related:      What are some recommended tools and methods of capacity planning for web sites and web-applications? Please feel free to describe different tools and techniques for different web-servers, frameworks, etc., as well as best-practices that apply to web servers in general. What is the difference between ubuntu-18.04-server and ubuntu-18.04-live-server? On the  there is a download link for ubuntu-18.04-live-server-amd64.iso, and on the  there is a download link for ubuntu-18.04-server-amd64.iso. Before 18.04, there is only a server version. What is the difference between these 2 images? Consider there is a box of mass $m$ at rest on the floor. Most books give an example that we need to do a work of $mgh$ to lift the box $h$ upward.  If we analyze this work done, the external force acting on the box by us should be equal to the weight of the box. Therefore the net force is zero which in turn there is no acceleration. If there is no acceleration and the initial velocity of the box is also zero, how can the box move upward? Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series What are good learning resources for a LaTeX beginner? Which book (free or otherwise) was the most useful to you when you started learning LaTeX?  I am frequently asked this question by friends who want to learn LaTeX, and I recommend the book which got me started, , but I feel that there might be better options around.  Also see . Can any one tell me how to prove that the set of rational numbers are countable? Prove give me a prove?  Thanks. So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. Identical coefficients estimated in Poisson vs Quasi-Poisson model I have a question about the following passage on pg. 89 of Zee's QFT in a nutshell:     At first sight, what Dirac wanted does not make sense. The equation is supposed to have the form "some linear combination of $\partial_\mu$ acting on some field $\psi$ is equal to some constant times the field." Denote the linear combination by $c^\mu\partial_\mu$. If the $c^\mu$'s are four ordinary numbers, then the four-vector $c^\mu$ defines some direction and the equation cannot be Lorentz invariant.   The simplest equation I can construct that he is referring to is  $$(c^{\mu}\partial_{\mu}-m)\phi(x)=0, $$  for $m&gt;0$. Taking $\phi(x)\rightarrow\phi(\Lambda^{-1}x)$, and $c^{\mu}\rightarrow\Lambda^{\mu}_{\ \ \nu}c^{\nu}$. Then, the left term transforms like   \begin{align*} \Lambda^{\mu}_{\ \ \nu}c^{\nu}(\Lambda^{-1})^{\sigma}_{\ \ \mu}\partial_{\sigma}\phi(\Lambda^{-1}x) &amp;= \delta^{\sigma}_{\ \ \nu}c^{\nu}\partial_{\sigma}\phi(\Lambda^{-1}x)\\ &amp;=c^{\sigma}\partial_{\sigma}\phi(\Lambda^{-1}x). \end{align*}  Then, we have that   $$ (c^{\sigma}\partial_{\sigma}-m)\phi(\Lambda^{-1}x)=0.$$  But doesn't this show that the equation is Lorentz-invariant? Make voting activity on your posts available on site metas (what you would get from rep if metas had rep) Child metas don't have rep, so there's no point in tracking a rep stat or reporting changes.  But one of the things rep updates do on main sites is to alert me when there's activity on older posts.  On metas, unless I revisit a question I'll never notice that there's been a bunch of voting (which might be accompanied by new comments that didn't ping me).  I'd like to know about that activity, not because of any ego reasons but because meta is where site policies get discussed and set and it's useful to know about changes in the community will.  A couple days ago I visited a meta post where, a few weeks ago, my answer had been voted down, and this time it was at the top.  Could we have something like the reputation-changes tab on meta?  I suspect that the data already exists, because I once  (and I see I'm not the only one).  If something akin to reputation is already being tracked internally on per-site metas, can we get it exposed? "Hail Mary", "Glory Be" in the plural Possible Duplicate:          In the rosary, a Catholic sacramental composed of prayer beads, there is a short prayer concluding each decade called "Glory Be" ("Glory be to the Father, Son and Holy Spirit...").  Would the plural be Glory Bes or Glory Be's?  Likewise, do you say 10 Hail Marys? The canonical way to return multiple values in languages that support it is often . Option: Using a tuple Consider this trivial example: def f(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return (y0, y1, y2)  However, this quickly gets problematic as the number of values returned increases. What if you want to return four or five values? Sure, you could keep tupling them, but it gets easy to forget which value is where. It's also rather ugly to unpack them wherever you want to receive them. Option: Using a dictionary The next logical step seems to be to introduce some sort of 'record notation'. In Python, the obvious way to do this is by means of a dict. Consider the following: def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return {'y0': y0, 'y1': y1 ,'y2': y2}  (Just to be clear, y0, y1, and y2 are just meant as abstract identifiers. As pointed out, in practice you'd use meaningful identifiers.) Now, we have a mechanism whereby we can project out a particular member of the returned object. For example, result['y0']  Option: Using a class However, there is another option. We could instead return a specialized structure. I've framed this in the context of Python, but I'm sure it applies to other languages as well. Indeed, if you were working in C this might very well be your only option. Here goes: class ReturnValue:   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return ReturnValue(y0, y1, y2)  In Python the previous two are perhaps very similar in terms of plumbing - after all { y0, y1, y2 } just end up being entries in the internal __dict__ of the ReturnValue. There is one additional feature provided by Python though for tiny objects, the __slots__ attribute. The class could be expressed as: class ReturnValue(object):   __slots__ = [&quot;y0&quot;, &quot;y1&quot;, &quot;y2&quot;]   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  From the :  The __slots__ declaration takes a sequence of instance variables and reserves just enough space in each instance to hold a value for each variable. Space is saved because __dict__ is not created for each instance.  Option: Using a  (Python 3.7+) Using Python 3.7's new dataclasses, return a class with automatically added special methods, typing and other useful tools: @dataclass class Returnvalue:     y0: int     y1: float     y3: int  def total_cost(x):     y0 = x + 1     y1 = x * 3     y2 = y0 ** y3     return ReturnValue(y0, y1, y2)  Option: Using a list Another suggestion which I'd overlooked comes from Bill the Lizard: def h(x):   result = [x + 1]   result.append(x * 3)   result.append(y0 ** y3)   return result  This is my least favorite method though. I suppose I'm tainted by exposure to Haskell, but the idea of mixed-type lists has always felt uncomfortable to me. In this particular example the list is -not- mixed type, but it conceivably could be. A list used in this way really doesn't gain anything with respect to the tuple as far as I can tell. The only real difference between lists and tuples in Python is that lists are , whereas tuples are not. I personally tend to carry over the conventions from functional programming: use lists for any number of elements of the same type, and tuples for a fixed number of elements of predetermined types. Question After the lengthy preamble, comes the inevitable question. Which method (do you think) is best? Any nice, stable ways to keep a window 'Always on top' on the Mac? I am looking for a stable way to keep any window 'always on top' on OSX.  , an app that does that requires SIMBL/easySIMBL, which it seems is not very stable.  Are there any better solutions? It's actually surprising that there is no native way to implement this. Problem using atan in pgfplots Official ESTA application website and are the others scams? A quick search for ESTA application returned many results. According to , the official site to apply for an ESTA is .  Are the others scams, legitimate third party resellers or "added value" services?    (defunct - domain is for sale circa 2020)     Can anyone confirm either way?  Updated 2020: The above have better "not affiliated with the U.S. Government" warnings now, but beware there are newer more sneaky sites starting all the time:        And fortunately more official guidance about this issue: Is ∅ equivalent to {∅}? I think they are, but I am not sure? If anyone could clarify, that would be great. Thank you! How to align prices in columns when both 2 digit and 4 digit variations of a price exist? I am trying to display data similar to coinmarketcap with prices for coins, some above 1$ in which case they are rounded to 2 zeroes, some below 1$ in which case they have 4 digits following the decimal. I saw a bunch of posts here where people have suggested to use right alignment for numeric applications/prices  Approach 1 Decimal aligned   1. $13,700.05  2.      $0.0048   Approach 2 Right aligned   1. $13700.05  2.   $0.0048   Which of these approaches do you suggest? Thank you for your time GPU Rendering causing strange lines When I render using my r9 390 graphics card, weird lines are appearing over the render. This problem has just started happening recently, as a few days ago I was rendering the same scene without the lines appearing. I don't believe I've changed any settings in Blender and the below scene is from a new file, so render settings should be default.   Rendering using my CPU is fine:   The only thing I can think that has changed is I recently updated my card to the latest drivers. Playing games with it is still fine and the card is quite new so I wouldn't think it's a problem with the card itself. $X$ and $Y$ are independent $U(0,1)$ random variables.   I have to find $E[X|X&gt;Y]$.   I found it out by integrating $\int_0^1 \int_y^1 x dxdy$ to get $\frac{1}{3}$ as the answer but this is wrong according to the answer key. What is the mistake that I have made? Radius of circumscribed circle of triangle as function of the sides Given the length ot the sides $a , b$ and $c$  of $ \triangle ABC$.  What is the length of the radius of the circumcribed circle?  After some formula substitution I came to the monster formula:  $$ \frac {a b c}{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}} $$  Can this formula be simplified?