So I'm studying for my course of linear algebra and the following problem was inside the book with exercises. &quot;Given a matrix $A \in \mathbb{R}^{n \times n}$, where $a_{ij} = i + j$, calculate it's determinant&quot;. The first thing I did was make a small visualization of what this sort of matrix would look like. I came up with this: $$ A = \begin{pmatrix} 1 + 1 &amp; 1 + 2 &amp; ... &amp; 1 + n\\ 2 + 1 &amp; 2 + 2 &amp; ... &amp; 2 + n\\ ... &amp; ... &amp; &amp; ...\\ n + 1 &amp; n + 2 &amp; ... &amp; n + n \end{pmatrix}$$ I set of to set up an expression for the determinant using cofactor expansion (I don't know if this is the correct English terminology for it, the formula is given below, I also found on wikipedia this is called &quot;Laplace Expansion&quot;?) to the first row (thus making $j = 1$ in the expression): $$ \sum^{n}_{i = 1}(-1)^{(i + 1)}a_{i1}det(M_{i1})$$ where $det(M_{i1})$ is the minor for the $i1$-th element. This is the point where I am stuck and could definitely use some help. I've tried writing $A$ in terms of summations to maybe see something in it, but this evolved into some very complex expression which I'm not even going to bother typing down (it's really, really long). I have a feeling that I can bring the $a_{i1}$ out of the summation but I'm not sure how I would go on doing that. Thanks in advance
Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$.  My work : I noticed that A is symmetric . Hence all of its eigen vectors are real .. That is all i have got .   Your help will be highly appreciated .Thank you .
Get list of ALL offers from Amazon Product Advertising API $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues I'm trying to find $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx$ where $\alpha&gt;0$ is real.  My approach was to take an integral along the real line from $1/R$ to $R$, around the circle counterclockwise to $-R$, along the real line to $-1/R$, and then around the circle clockwise to $1/R$.  I have encountered 2 problems with this:   This path encloses one pole, at $z=\alpha i$.  I found the residue at $z=\alpha i$ to be $\frac{\ln(\alpha)+i\pi/2}{2\alpha i}$.  However, this gives me that $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx=\frac{\pi(\ln(\alpha)+i\pi/2)}{2\alpha}$.  Since I have a real function integrated over the real line, there cannot be an imaginary part.  Where did I go wrong?  (Also, doing a few examples, the correct answer seems to be $\frac{\pi\ln(\alpha)}{2\alpha}$, the same as I have but without the imaginary part.) At first chose my path so instead of going all the way around the upper semicircle, it only went 3/4 of the way around, as I wanted to avoid anything that might go wrong with the discontinuity of $\log(x)$ at the negative real axis.  When I do this, though, I get a different answer than before (the denominator of the fraction is $\alpha(1-e^{3\pi i/4})$ instead of $2\alpha$.  What am I doing wrong that gives me different answers? wpf combobox binding How to set up beamer to grey out past items but hide future items? If $\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1$, then what's the value of $4\left(\frac{\cos^3y}{\cos x}+\frac{\sin^3y}{\sin x}\right)$? Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$ Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$. It is the distance of a point in a triangle from its vertexes which gives the minimum when $(x,y)$ is on the Fermat's point.  But now I am stuck in finding the location of Fermat's point.  $(1) \quad 4$ $(2) \quad 5$ $(3) \quad \sqrt{3}+2\sqrt{5}$ $(4) \quad 4\sqrt{2}$ I can show this best by a picture:    This cross also shows when I export it as raster or pdf. What is it and how can I get rid of it?  I am using Qgis 2.4 and the map was done with the help of the Grass plugin AniMove on Linux Mint. Count and use the number of items in advance Getting started with neural networks for forecasting I would like to be able to transpose a tabular(x) table like in the following example. It is not necessary to be completely automated, but I don't want to change the data order, which is a painful and stupid. By "not completely automated" I mean that e.g. the headers can be done seperately etc.  Let's see the MWE:  \documentclass{article}  \usepackage{booktabs} \def\Midrule{\midrule[\heavyrulewidth]}  \begin{document}  \centering  \begin{tabular}{*6c}\toprule \bfseries Name &amp; Alice &amp; Bob &amp; Chuck &amp; Dave &amp; Eve\\\Midrule \bfseries Sex &amp; Female &amp; Male &amp; Male &amp; Male &amp; Female\\\midrule \bfseries Age &amp; 18 &amp; 19 &amp; 20 &amp; 21 &amp; 22\\\bottomrule \end{tabular}  \bigskip  Which I want to look like  \bigskip  \begin{tabular}{*3c}\toprule \bfseries Name &amp; \bfseries Sex &amp; \bfseries Age\\\Midrule Alice &amp; Female &amp; 18\\\midrule Bob &amp; Male &amp; 19\\\midrule Chuck &amp; Male &amp; 20\\\midrule Dave &amp; Male &amp; 21\\\midrule Eve &amp; Female &amp; 22\\\bottomrule \end{tabular}   \end{document}     Note: I know that , but I would like to get it without expl3 (mostly because people would might use my code quite often have an old LaTeX installed and I cannot be convincing them to change this). The vote fraud script breaks legitimate polling questions - A serial downvoter's lament I've downvoted several of the answers on these Meta discussions:       Both questions are typical self evaluations, Stack Exchange runs them on all young Beta sites and as you may have noticed they are essentially polls, a series of example questions from the site that we are supposed to vote on.  Today I noticed that almost all of my votes have disappeared on both evaluations, only an upvote remains in the later one. My votes triggered the serial voting script, something that :      Heh, it appears I just serially downvoted you (I downvoted a lot of the answers here), hopefully the script is smart enough to not reverse my votes. Will check back in 24h, and if it does, I'll rant about it on MSO ;)    My voting behaviour on both evaluations was very simple:   I had the evaluation open in the left monitor, Opened all the example questions in their own tab, in the right monitor, Scanned each question quickly, and voted on the Meta answer (most of my votes were downvotes), Repeat for the next one, and so on...   I didn't spend more than a few seconds on any question, as I'm fairly regular on the site and I've already read all of them while casually browsing the site. In fact, if my memory isn't failing me, I had voted (up/down) and/or commented, and/or voted to close on each of the 20 example questions. I might have spend a minute or two re-reading a couple, but I mostly scanned through them. In any case I think my voting pattern is quite normal, and the script should be somehow tweaked to take serial posts, posted by the same user in quick succession, into consideration, especially on Metas.   Since I had anticipated that the script would kick in, I could have tried to fool it by pausing a bit between votes or trying to look for other Meta posts to vote on, in between voting on the evaluation answers. However I'm a very active voter on Meta Workplace, there aren't many posts I haven't already voted on, and the few that I haven't is probably because I neither agree or disagree with (my opinion on them is simply "meh"). In any case I think my voting behaviour was natural, and although the serial voting script is awesome, this time it caught and reversed a perfectly legitimate behaviour.  Animuson, in , mentions that the script isn't really optimal on Area51 as well:      Yes, please implement this on Area 51 proposals too! Oftentimes a single user would suggest a plethora of "example" questions to their own proposal which are just terrible, yet all of everyone's votes get reversed and the question just sits there at 0. I've noticed several times that my votes have gotten reversed.    I haven't noticed it myself, but it certainly seems plausible, especially early in a proposal's life.  Mysticial found a related discussion, , but it's focused on voting patterns on main sites, where reputation is also a factor. I can understand jumping through hoops to satisfy the script on main sites where, unfortunately, there are a lot of people trying to get some cheap rep or "punish" others with their downvotes, but not on Metas, and especially not on these evaluation posts.   Voting fraud is a problem, and more often than not the script works wonderfully, but in my case it was extremely counterproductive. Critical evaluations are essential for a young Beta site, and their answers typically only get a handful of votes. I wonder how many of these votes have silently gone away...  Fix this please. Initial connection to SQL Server Connection Is Slow. Why? 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I know there are several answers scattered around this site, but they are either hard to find, just cover a specific device, or don't cover all steps I finally needed. So I take this as a chance for a canonical, device-independent question and give it a detailed answer below:  How can I get my Android device seen and used by adb on Linux? How do you install an APK file in the Android emulator? By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this  Edit: By using Hans Lundmark hint, I get:  $$(x^2+1)^2-x^2$$   Is it fully factorized? What's the difference between "these" and "those"? First of all, I'm not a native English speaker, but in school I learned that these is used if referring to something near, and those is used when referring to something far away (temporally or locally). But now I'm sometimes watching English movies and notice sentences like "Have you seen those boots?" while the boots were only centimeters away. Is this just not correct, or have I learned the wrong rules?  Another question is, what if I'm referring to something abstract such as ideas or thoughts so that I can't say if they are far away or not. Should I use those or these?  Edit  So it can't be really wrong in any case if I use these respectively those? Why is a black hole black?