A real $2 \times 2$ matrix $M$ such that   $$  M^2  = \begin{pmatrix}  -1 &amp; 0 &amp; \\   0 &amp; -1-\epsilon  \end{pmatrix} $$   exists         for every $\epsilon &gt; 0$,   for at least one $\epsilon &gt; 0$,   for no $\epsilon &gt; 0$,   none of the above.      I have no idea on how to do this question. Hints?  Thanks
A real $2 \times 2 $ matrix $M$ such that $$M^2 =  \begin{pmatrix} -1&amp;0 \\ 0&amp;-1-\epsilon  \\   \end{pmatrix}$$ (a) exists for all $\epsilon &gt; 0$. (b) does not exist for any $\epsilon &gt; 0$. (c) exists for some $\epsilon &gt; 0$. (d) None of the above is true.  Attempt: I couldn't think of any theory base to prove / disprove the existence of such a matrix. Could someone please give me a hint on how to go about this problem? Thank you very much for your help in this regard.
May I use Google Fonts for creating a logo with commercial purposes? I want to create a logo for a contest, the winner will be paid and the winning logo will be the institutional logo of an academic entity, so I need a font which can be used freely or at a cheap cost for commercial purposes.  I see that Google Fonts are delivered as "free fonts" for websites:  May I use them freely for graphic design products (a logo in particular) as well? I just want to write \Sha without ruining everything I realize similar questions have been asked before but I am not satisfied with any of the answers I've seen. I would like to use the Cyrillic letter Ш to denote a particular group in mathematics. Most solutions involve using OT2 or T2A encoding; for example, at . However, this makes the other text in the document look bad.   For example, using the solution above, I get    while I would like to have     The first seems to be shaded strangely and the characters don't seem to align right.  Does anyone know how I can write the character Ш (in math mode) without changing the default font for the rest of the text? Is there any way to keep a command from being added to your history? I have a command that I want to keep out of my history file, and I really don't care to have it there when I search the history stored in memory, though that's less of a concern. Is there any way to prevent this, or do I just have to go back and edit my history file.  update: I didn't realize this might be shell-specific. My shell is zsh. You're welcome to answer for other shells so people know how to do this in their shell. How do I view 'git diff' output with my preferred diff tool/ viewer? When I type git diff, I want to view the output with my visual diff tool of choice (SourceGear "diffmerge" on Windows). How do I configure git to do this? In a script I'm trying to find the Terminal.app window containing a specific tty device and raise that window to the top. Here's what I have in Applescript so far:  tell application "System Events"   set foundWin to false   if (name of processes) contains "Terminal" then     tell application "Terminal"       set theTabs to first tab of every window where tty is "$(tty)"       repeat with theTab in theTabs         if class of theTab is tab then           set theWin to (first window whose tabs contains theTab)           set selected tab of theWin to theTab           set foundWin to true         end if       end repeat     end tell   end if   if foundWin then     --RAISE THE WINDOW?!   end if end tell   Where I'm getting stuck is the "raising the window" part.  Here are some things that are not quite what I want:  set frontmost of theWin to true -- this brings the window to the front of the group of Terminal.app windows but doesn't raise it above any other windows.  tell application "Terminal" to activate -- this brings every Terminal window to the front in a big stack. I just want the one window.  tell application "System Events"     set theSysWin to first window of process "Terminal" whose name is (name of theWin)     perform action "AXRaise" of theSysWin end tell   This almost does it, but what it does is to raise the terminal window to the #2 position, still underneath the active window (if the active app is something other than Terminal.app.)  click theSysWin at {10,50} -- doesn't seem to do anything.  click first static text of theSysWin -- doesn't seem to do anything.  Is there a way to do this? It doesn't have to be in Applescript.  EDIT I did find this web page (  ) quoting an Obj-C/Cocoa call:  SetFrontProcessWithOptions(&amp;psn, kSetFrontProcessFrontWindowOnly);  I'm not familiar with either ObjC or Cocoa, and this is (ultimately) getting invoked from a shell script, so not sure where to go from there. Comparing signatures of apks I have two .apk files that are supposed to be different versions of the same app. What is the best way to check if they where both signed with the same key?  I want to make sure no mistakes were done and the newer version would be treated as an update of the same app. what is a thyristor As part of my 2330 lv3 am required to know about thyristors. I just want to confirm my understanding is correct. A thyristor is basically similar to a transistor, only it stays switched on when it's trigged for as long as there isn't a break in the circuit it's switching. Am I correct? Run a program X times Concatenate multiple rows I'm currently building a town in Minecraft Pocket Edition and I'm wondering how many blocks long and how many blocks wide the world is. Prove continuous functions $f$ and $g$ are equal I am creating an animated sprite list in Adobe Illustrator. I finished creating the sprites and used the Horizontal/Vertical Distribute Center commands to distribute the objects equally in some area.  But, if the objects are spread too far apart, the Distribute Center tool leaves, albeit equal, much distance between the objects. The opposite is true: if the objects are too close then distributing them will just overlap them.  My question is, is there any tool that just puts objects side by side?  I imagine that this tool would work if the objects aren't even equal in size.   I have viewed  and answer but it is not what I need. SQL Server CE in a multi user scenario - is this really, really stupid? Right now, I have everything in a partition (except EFI system partition). How can I split the partition into two, one for / and one for /home?   How would you use parted to do that step by step?  How would you use gparted to do that step by step?  How would you use LVM to do that step by step?   One thing I am not sure of is that I can't just shrink the partition and create a second partition on the space that is freed by shrinking, because that doesn't move /home to the new partition.  Thanks.    $ sudo parted -l [sudo] password for t:  Model: ATA TOSHIBA MQ01ABF0 (scsi) Disk /dev/sda: 500GB Sector size (logical/physical): 512B/512B Partition Table: gpt Disk Flags:   Number  Start   End    Size   File system  Name                  Flags  1      1049kB  538MB  537MB  fat32        EFI System Partition  boot, esp  2      538MB   500GB  500GB  ext4  $ sudo df -h [sudo] password for t:  Filesystem      Size  Used Avail Use% Mounted on udev            1.7G     0  1.7G   0% /dev tmpfs           340M  1.4M  338M   1% /run /dev/sda2       457G  6.7G  428G   2% / tmpfs           1.7G   27M  1.7G   2% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           1.7G     0  1.7G   0% /sys/fs/cgroup /dev/sda1       511M  6.1M  505M   2% /boot/efi tmpfs           340M   20K  340M   1% /run/user/1000 tmpfs           340M     0  340M   0% /run/user/106 A coin is tossed $n$ times. What is the probability of getting odd number of heads?   I started this chapter sometimes ago and faced in front of a tough problem. At first I started considering cases. Case-I : The probability of getting 1 head.Case-II : The probability of getting 3 head and so on. But there are many cases. So how can I solved this . Please help me. Thank you! Does the English language have an official Academy? For some languages, there are academies that decide topics such as grammar and spelling of things, for example, for the Spanish language, there are 22 academies in 22 different countries, all making decisions on spanish grammar in their country. I would like to know if there is something that serves the same purpose for the English language. Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$ It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$  Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$  I was thinking of making polar transformations $X=r \cos \theta, Y=r \sin \theta$  Then I am getting stuck at ranges of $\theta$ How can I depict vehicle traffic during daytime in a long exposure without washing out the vehicles? As a result, I am able to be understanding and accommodating to the people that I work with, including clients, which I believe is an important asset for a professional to have.   I wanted to make sure it is absolutely known that I mean to acknowledge clients as a class of people I can work with. Is this the right way to include it with commas? Update on Ubuntu - coloured lines around everything Everyday I check for updates on Ubuntu 16.04 LTS. I updated to 4.4.0-65-generic today and now on everything I'm getting thick rainbow coloured lines around every dialog box in every program. My setup is pretty standard. Skylake i5-6500, Nvidia 950gtx. Does anyone know how to fix it?