Let $A, B \in M_n(\mathbb{R})$ such that : $AB-BA$ invertible and $A^2+B^2 = AB$ then prove that : $3 \mid n$.   I tried to manipulate or find some factorisation in order to be able to use efficiently eigenvalues...  but I dind't find anything.   Thank you !
The question is:     Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular?   From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this?  Edit:  $$A^2+B^2=AB\implies\\ A(A^2+B^2)=A^2B \implies\\ A^3+AB^2=A^2B \implies \\ A^3+(A^2+B^2)B=A^2B \implies \\  A^3+A^2B+B^3=A^2B \implies \\ A^3+B^3=0 $$
How does polynomial long division work? Is there a $(6,9,3)_2$ code? Does there exist a $1$-error-correcting binary code with block length $6$ and $9$ codewords?   The Hamming bound says that for any code $C$ with those parameters, $|C| \le \frac{2^6}{1+6} \approx 9.14$. So, we can't rule out the existence of such a code using the Hamming bound. The Singleton bound says that for any code $C$ with those parameters, $|C| \le 2^{6-3+1} = 16$, so we can't rule out the existence of such a code using the Singleton bound. Also, this can't be a linear code, since $\log_2 9$ isn't an integer. That rules out the possibility of just enumerating the possible generator matrices.   I feel a bit silly asking this, but the direct approach (assume $000\ 000$ is in the code, then all others must have Hamming weight $\ge 3$, ...) becomes unmanageable quickly. How else can I go about this? I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help! When getting a new credit card, what is the policy in regards to signing the strip on the back? Should I sign it? Should I write "See ID"? Should I leave it blank? Should I sign it AND write "See ID"?   I see all sorts of different advice but it seems like it's all based on urban legends, hearsay, or self interest (merchants and banks seem to advise you to do what's best for them, not what's best for the customer).  My question: For an average everyday consumer who uses a card fairly frequently, what is the best practice? What practice is most secure? Small caps font warning in beamer using biblatex Consider the following MWE  \documentclass{beamer} \usepackage{silence} \WarningFilter{biblatex}{Patching footnotes failed}  \usepackage{lmodern} \usepackage[T1]{fontenc}  \usepackage[style=authoryear-comp,backend=biber]{biblatex} \addbibresource{biblatex-examples.bib}  \begin{document} \begin{frame}   \fullcite{sarfraz} \end{frame} \end{document}   This generates the warning:  LaTeX Font Warning: Font shape `T1/lmss/m/sc' in size &lt;10.95&gt; not available (Font)              Font shape `T1/lmr/m/sc' tried instead on input line 28.   Is there a way to avoid the warning without using the silence package? Story where radically different cultures live together and are defined by rulesets, some choose not to see each other, a house is used as a spaceship I think this novel came out in the 2000s.  There are many cultures that live close together but are very different, for example some pretend to be pre-industrial.  One of the main characters is a youth who grew up in a pretend pre-industrial society and is now exploring the rest of the solar system? ring-world?   Each culture experiences their own reality that is defined by a ruleset.  One of the rulesets becomes aggressively self-propagating and this is the problem that drives the plot.  At some point a tumbling house, just a normal house, is used to travel across space. Could anyone explain when to use a noun or a gerund?  Are these sentences correct? What is the difference between usage of gerund and noun there?        I worked on the improvement of the article.   I worked on the article improving.   I worked on the improving of the article.                The investment idea was published.   Investing idea was published. Can't save settings after copying site to new server Image upload from photo gallery So this is pretty obvious. Image upload?    How about allowing to attach images in questions from the photo gallery, by perhaps sending the image for off-site storage like done from the full-web version? Rotate touchscreen and disable the touchpad on Yoga 2 Pro in rotated mode On my Lenovo Yoga 2 pro I installed Ubuntu 14.04 32bit and kept the installed Windows 8 on another partition.  In Windows, When you turn it into tablet mode, the keyboard and touchpad turns off so you don't accidently click on it at the back of the "tablet".  In ubuntu 14.04 only the keyboard turns off, but the touchpad stays active. Not even the Fn+F6 combination doesn't turn it off.  So far I can only disable it with  synclient TouchpadOff=1   (and re-enable with 0)  I tried xev to get the keycode for Fn+F6 but pressing this combo generates no output. Turning the monitor to the back neither.  How can I disable the touchpad automatically when I rotate or turn the monitor on the back, and re-enable the Fn+F6 Hotkey?      UPDATE: After some weeks sudo apt-get upgrade Fn+F6 is working now, so there is only the question how to rotate the screen and how to disable the touchpad automatically when rotating the screen. When writing answers on here, it's often necessary to create a new script and chmod +x it to make it executable. I won't go into whether this is a good thing or not, but explaining to people that they need to fire in three separate command (and teaching them to use nano!) is a bit of a waffle.  Is there anything easier? In IE 9 you can type in whatever number you want to get the perfect zoom but in Chrome all I see is the + and - in the drop menu.  How do I choose a custom zoom and set it as the default? I'm looking to build a guitar pedal switcher based on a Raspberry Pi or Arduino (have not yet decided, but I have an unused Raspberry around and I lean towards that). Anyway, digital logic.  Now, I need to switch audio signal obviously and I need to do it quietly (both physical silence and electrical silence). Please note that the signal is line level throughout most of the build, except for the input which is much lower. I will probably have 2 or 3 loops before the preamp so those may not encounter line level signal.  From what I read and know, so far, mechanical relays are not really an option due to the clicking noise they make. I also read something about FETs, but that's a bit out of my league and I would appreciate any info on these.  PS: I will probably use different power rails for the digital and analog paths.  PS2: The whole thing will be mounted in a 1U rack enclosure, so please tell me if I should take into consideration any thermal issues. I'm fairly new to DMing and so is my group, so we are all on a learning curve together. We're playing D&amp;D 5e.  Last week a fighter in our group tamed a wolf with Animal Handling and then requested that he keep the wolf. I allowed this, but now I'm not sure about how to play the wolf.  Specifically, how can he train the wolf to perform certain actions such as attacking?  What other things might I need to be aware of with a PC owning a "pet"? Why isn't Force Speed used more often? Delayed erasure experiment I find that often after awakening from suspend on Ubuntu GNOME 16.04 with GNOME 3.20 (I have only found this issue on the latest version, though I have not tried with GNOME 3.18 on 16.04) that I get this strange font issue which can't be fixed by restarting gnome-shell and seemingly only by restarting my entire machine:            I do not believe that this is a theme issue as I have tried all the built-in themes (including the dark and light versions) and it is the same for them all. Though it does not seem to affect gnome-shell areas, that is the top-bar, the Activities Overview, etc, are not affected for some reason. So what could this issue be caused by and how can I fix or workaround it?  Information Update:  I have had a look at  but the only options in the gnome-tweak-tool for Antialiasing are Rgba, None, and Grayscale, but none of them seem to have any affect on the issue. Is there something I'm missing? How to derive this binomial identity? We have given isosceles triangle $ABC$ with baseline $AB$. Point $M$ is midpoint of AB. Draw perpendicular line to side $AC$ through $M$ which intersects $AC$ in point $H$. Let $P$ be midpoint of $MH$. Show that lines $BH$ and $CP$ are perpendicular to each other.  I also have "hint": Construct height $BD$ to side $AC$ and look at the triangles $ABD$ and $MCH$.  Analytically there is no problem but I get stuck in geometrical approach to solution. As seen in this video: (Skip to 5:53)  When a player is slain, they drop an Enchanting Table on top of their inventory.  How was this achieved and how can I build a system (using command blocks) that does that? Come to think of it, what happens to a player's location when he/she dies? Does it reset to the spawnpoint? Stay the same (where they died) until they respawn? Or becomes a null value until they respawn?