If $$ A =         \begin{bmatrix}         3 &amp; 4 \\         4 &amp; -3           \end{bmatrix}$$  Can someone find $\mathbf e^A$ ?  Edit 1:- I got this question in my fucntional analysis paper ! How is it even related to functional analysis ? Can somebody explain ?
For a 2x2 matrix A satisfying $A^k=I$, compute $e^A$  Oh, the exponential of a matrix is: $e^A=\sum_{i=0}^\infty\frac{1}{i!}A^i$    I thought I'd solved the $e^A$ form but I actually did something really silly, now I'm a little stuck.  If $A^k=I$ then $A^{k+1}=A$ and we have a cycle forming, so we will get:  $$\sum^{k-1}_{i=0}\sum^{\infty}_{j=1}\frac{1}{(kj+i)!}A^i$$  (Or something like this form, I don't have paper to hand and just spotted that now)  Is this what the question wants? It doesn't use the 2x2 property.  I believe the answer lies in finding an expression for the inner summation, I can do this for two terms (think of $e+e^{-1}$ all the odd power leave)
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime. Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.  Can someone help me with this? 16.04 new installation gives grub-efi-amd64-signed failed installation /target/ ubuntu 16.04 at the end I am installing Ubuntu 16.04 in an Intel NUC computer with UEFI enabled. I am using an SSD from another computer. During the installation I chose Erase the disk and make a new full install  At the end of the process I see:  grub-efi-amd64-signed failed installation /target/ Ubuntu 16.04    And the system does not boot.  I tried to use boot-repair but apparently it does not solve the problem. Finding the distribution of $\|X-\mu \|_\Sigma^2$ with $X \sim N(\mu,\Sigma)$ Let $\Sigma$ be positive definite and symmetric and let $\| x \|_\Sigma := \sqrt{x^t\Sigma^{-1}x}$ be the Mahalanobis norm. Let further $X \sim N(\mu,\Sigma)$ be a $p-$variat normally distributed random vector. Find the distribution of $\| X-\mu \|_\Sigma^2$.   I do not kow how I could do that. Could you help me? Reading over the Monk I saw that for Martial Arts:     When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.   This implies that non-monk weapons do not count towards being able to make this single unarmed strike. However, when looking over Flurry of Blows:     Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.   The requirement doesn't seem to be there anymore, and I didn't see where this discrepancy was covered in . This  says that you can use flurry of blows as a bonus action on any form of attack.   So does the limitation really only exist on a single unarmed strike, but you can use whatever attack action you want to trigger Flurry of Blows? Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$? Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? (Here $k$ is a field.)    Thanks for any help. Application is "Ready to Sale" but not reflected on iTunes Store Sophix tablet will not work without an SD card My Sophix tablet was working without the SD card, but now the camera will not work. I get the following error message:     Insert a SD card before using. What is the standard/recommended font to use in papers? I looked around but did not find that anyone has asked this before, but what are the fonts that are standard/recommended while writing academic reports/papers? In English, words with a 'g' followed by a front vowel (e, i, y) can be pronounced with a soft g or a hard g:   Words with Germanic roots are usually pronounced with a hard g: gear, get, gift, give Words with Latin and Greek roots are usually pronounced with a soft g: gem, general, giraffe, giant   But how should a purely English word (if such thing even exists) with a 'g' followed by a front vowel be pronounced? In other words, if an English speaker saw a new word of unknown origin (eg: a neologism) that starts with gi- or ge-, how would they pronounce it? How to show that Klein four-group is a normal subgroup of the alternating group $A_4$ Adding x and y values while creating a label QGIS I'm trying to create a label on a map in QGIS.  I can open the setting where there are an X and Y button and a few other properties, like opacity and distance.  I'm pretty sure the problem is that I need to enter the proper X and Y coordinate here, but I don't know how to get to where it lets you enter the coordinates. Cannot install a version of GDAL via Anaconda that permits reading BigTIFFs Is there any way to automatically rename accessor/mutator when a variable they get/set gets refactored -> renamed (Eclipse 3.4)? Convert a line drawing from raster to vector I have a raster image, basically a scanned drawing of the outlines (and some detail) of various objects, and I wish to convert these lines to vectors, using either Illustrator or Inkscape.  My present solution is to have the raster image as a background layer, and in a new layer above it I use the pen to follow the lines as close as possible.  Later I tweak these lines to fit the raster image better. That has worked fine for simple drawings. However for some of my extremely complex drawings, this is very tedious to do.  Is there any tool or filter of some sort that is capable of recognizing the lines in line drawings and creating vector paths from them? How can I add a newline in the text of a label in WPF such as the following?  &lt;Label&gt;Lorem    ipsum&lt;/Label&gt; Antecedent Precedence? "There is/are more than one". What's the difference? While adding to an Answer to  question, I needed to use the above phrase, and I suddenly realised I was unsure whether to write "is" or "are".   There is more than one way to skin a cat. If there are more than one species of cat, we will flay each species differently.   I don't think that second example above sounds quite right to me (disregarding the meaning - sorry!), but in my original Answer I felt "is" would have been even worse.  Is "are" always wrong here? If so, what exactly is the rationale? If not, is there a case where it's unquestionably preferred – and why is that?    Note that in both the problematic Answer and my example attempting to mimic the same context more briefly, the writer doesn't know how many there are. There may be one, more, or even none.  Getting even more specific, the writer might have an opinion on how many there might be, and wish to convey his leanings by choice of verb form if language allows this to be done succinctly. We are always introduced to magnets by pictures illustrating a parallepiped magnet, a horseshoe magnet, etc... Anyway the first time we actually calculate a magnetic field is for eelectromagnets. Indeed I've rarely seen explaining or calculating the magnetic field of a permanent magnet. I know formulas for the magnetic field of a moving charge, around a conductive wire, inside a solenoid, but I don't have idea on how to go about calcualting the filed of a simple permanent magnet. It's like that without the help of electricity I'm stuck. I'd guess it would be heavily dependent on the magnetic material, its shape and the medium in which is placed. I hope that someone would provide me some insights on why such formulas are  never introduced in most physics courses and lay down some examples of a permanent magnet field caluclation. I need to prove that $H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.  My proof to the $(\Rightarrow)$ direction seems too much trivial:  Let us assume there exist $A$ so that $A/H\lhd G/H$. Then by definion, $H$ must be normal in $A$. Because $H$ is maximal, we get $H=A$ and therefore $A/H={1}$  Is it correct?  Update:  Now I see that I need to prove that not only $A\lhd H$ but also $A\lhd G$. Assumig I have proven that, is the proof correct? is it possible to redefine { and } in C?