$A=\begin{bmatrix}3 &amp; -2 &amp; 5\\ 1 &amp; 0 &amp; 7\\ 0 &amp; 0 &amp; 2\end{bmatrix}$, Find the eigenvalues of A.   I realized that if I swap columns I and II then I can make it an upper triangular matrix. Then the detrminant would be the product of the elements of the main diagonal. And then I would just need to find the roots of that.  However I know that swapping columns flips the sign of the determinant, but I don't know how that will effect finding the eigenvalues.  So I tried it anyways and got determinant of $(x+2)(x-1)(x-2)$ which has roots -2, 1, and 2. But I know that this is incorrect because the answers are supposed to be $\lambda=1,2,2$.  What did I do wrong?
If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix?  NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of the specific examples that come up can be marked as duplicates of this post. See .
How can I find out what is using the hard disk? Possible Duplicates:             Once or twice a day my computer starts writing significantly hard to the hard drive, the HD led is ON and the performance hit is noticeable. A strange observation is that the Task Viewer does not show I/O activity - even Process Explorer shows no CPU activity and no I/O bytes read or written.  Is there a way to know which application is using the hard drive and how much data is it reading or writing? Proof for triangle inequality for vectors Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below.  By considering $u$ and $v$ as two sides of a triangle, we can see that the lengths of the third side is $\| u + v \|$ and we have $\| u + v \| \leq \|u\| + \|v\|$.  Under what circumstance equality occurs and how can one prove that? Prove that the minimum number of cycles is $m-n+1$ The question I have is:   Prove that the minimum number of cycles is $m-n+1$ in a connected graph. Where one cycle is a path that starts that begins and ends at the same vertex. Where $m$ is edges and $n$ is the vertices.   I have no idea where to start. A few hints would be appreciated. Please do not provide me with the answer. My ubuntu partition was deleted and I can't boot from either a DVD or USB How to make sure an application keeps running on Linux I'm trying to ensure a script remains running on a development server. It collates stats and provides a web service so it's supposed to persist, yet a few times a day, it dies off for unknown reasons. When we notice we just launch it again, but it's a pain in the rear and some users don't have permission (or the knowhow) to launch it up.  The programmer in me wants to spend a few hours getting to the bottom of the problem but the busy person in me thinks there must be an easy way to detect if an app is not running, and launch it again.  I know I could cron-script ps through grep:  ps -A | grep appname   But again, that's another hour of my life wasted on doing something that must already exist... Is there not a pre-made app that I can pass an executable (optionally with arguments) and that will keep a process running indefinitely?  In case it makes any difference, it's Ubuntu. What is the largest inductance value ever attained(in Henries)? Is there a way to force Google Chrome to remember passwords for sites that have autocomplete turned off? Clarification: I am a programmer, but ideally, I'd like a method that I could help regular users implement, too. Defining the appropriate Projected Coordinate System Basically I have a setup like the following:  Array (  [0] =&gt; Array ( [0] =&gt; stdClass Object ( [nid] =&gt; 1 [title] =&gt; title1 [uid] =&gt; 1 [parent] =&gt; 0 [weight] =&gt; -15 [name] =&gt; name1 [value] =&gt; 0 )                [1] =&gt; stdClass Object ( [nid] =&gt; 2 [title] =&gt; title2 [uid] =&gt; 1 [parent] =&gt; 0 [weight] =&gt; -7 [name] =&gt; name2 [value] =&gt; 100 )                [2] =&gt; stdClass Object ( [nid] =&gt; 3 [title] =&gt; title3 [uid] =&gt; 2 [parent] =&gt; 0 [weight] =&gt; -1 [name] =&gt; name3 [value] =&gt; 0 )                [3] =&gt; stdClass Object ( [nid] =&gt; 4 [title] =&gt; title4 [uid] =&gt; 2 [parent] =&gt; 0 [weight] =&gt; 1 [name] =&gt; name4 [value] =&gt; 80 )               )   )   What I need is a way to sort all the arrays inside the parent array by the [value] key in the Object. I've been trying for about 2 days now with usort and different methods but I just can't seem to get my head around it. The [value] key will range anywhere from 0 to 100 and I need all of the arrays sorted in decreasing order (IE: 100 down to 0). I am planning a trip to Italy this coming January. My plan is to land by plane in Roma and then go to France by train/bus stopping in some Northern Italian cities on the way, enjoying the architecture, the atmosphere, the art and and the food.   I want to do it at my own pace, without heavy scheduling. Ideally, I would just decide which cities to visit, and take a train whenever I feel it. I often do this here in South Korea where I am living, taking advantage of the "standing seats" as it is called here, that is travelling (for a cheap price and no reservation) standing on the romm between two cars. From my experience, it is also doable in Taiwan (at least in some trains).  In France, regional trains usually does not need any reservation, so you can buy a ticket and take the train whenever you want.  Can I expect a similar system in Italy, that is a way to make short travels between close cities without reservation?   Example of "short travel": Pisa - Firenze. How can I prevent Overflow for the Gamma function? I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation  $p\left(s|\alpha\right) = \sum\limits_{i=1}^O\log\left(\frac{n_i!}{\prod\limits_{j=1}^kn_{ij}!}\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)} \prod\limits_{j=1}^k \frac{\Gamma\left(n_{ij}+\alpha_j\right)}{\Gamma\left(\alpha_j\right)}\right)$  I have a python script to calculate the log likelihood. It works for small values, but when $n_{ij}$ gets high $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ python runs into overflow error (the max for math.gamma is between 171 and 172).   Is there a way to change the likelihood function so that $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ don't run into overflow errors. Can I just divide the values in $\Gamma$ by for example 100, if I then compare the likelihood between $p(s|\alpha)$ with different $\alpha$ values do the best $\alpha$ values still have the highest likelihood? How to avoid the pop up message while downloading an application (.exe) using selenium I am trying to download an application and check if it's downloaded into my local system or not. But when I click the download link I have a popup with Save or Cancel options, which is stopping me from downloading the file. I googled and found some solutions and tried to change the profile of Firefox by adding the below code, but it is not working.  firefoxProfile.setPreference("browser.download.folderList", 2); firefoxProfile.setPreference("browser.download.alertOnEXEOpen", false); firefoxProfile.setPreference("browser.helperApps.neverAsksaveToDisk", "application/x-msexcel,application/excel,application/x-excel,application/excel,application/x-excel,application/excel,application/vnd.ms-excel,application/x-excel,application/x-msexcel"); firefoxProfile.setPreference("browser.download.manager.showWhenStarting", false); firefoxProfile.setPreference("browser.download.manager.focusWhenStarting", false); firefoxProfile.setPreference("browser.helperApps.alwaysAsk.force", false); firefoxProfile.setPreference("browser.download.manager.alertOnEXEOpen", false); firefoxProfile.setPreference("browser.download.manager.closeWhenDone", false); firefoxProfile.setPreference("browser.download.manager.showAlertOnComplete", false); firefoxProfile.setPreference("browser.download.manager.useWindow", false); firefoxProfile.setPreference("browser.download.manager.showWhenStarting", false); firefoxProfile.setPreference("services.sync.prefs.sync.browser.download.manager.showWhenStarting", false);   This is the code I added for setting up Firefox preferences, but I still get the popup. Is SO/SE about to be sold (and does that explain recent events)? Best way to find all factors of a given number Can a PC with the Alert feat alert his friends? How can I verify the following statement?     The image of a morphism $f : A \longrightarrow B$ is defined as $\text{Im}(f) = \text{Ke}r(\text{coke}r f)$ whenever it exists (e.g., in every abelian category). The morphism $f : A \longrightarrow B$ factors   uniquely through $\text{Im} f \longrightarrow B$ whenever $\text{Im} f$ exists, and $A \longrightarrow \text{Im} f$ is an epimorphism and a cokernel of ker $f \longrightarrow A$ in every abelian category. How do I programmatically log in a user with a POST request? Do anyone have a working REST Login on Drupal 8? This is what I tried. POST /user/login HTTP/1.1 Host: 8.d8.local Content-Type: application/x-www-form-urlencoded Accept-Encoding: http://nikhilmohan.in Cache-Control: no-cache  name=test&amp;pass=password&amp;form_id=user_login_form  It returns me HTML instead of JSON. It is a general concept that horizontal forces can only influence physical quantities along horizontal direction; similarly vertical force can produce acceleration etc. only along vertical direction. This is the same concept that helps us study projectile motion as a combination of two independent motions. If this relation is universally true, how come, then friction depend on a force which normal to it? Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$ About a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$  If we'll take a line $f(x)=\begin{cases} ax &amp;, x\in\mathbb R\\ 0 &amp;, x=\infty  \end{cases}$ then we'll have a bijection.   But about this function: $g(x)=\begin{cases} x &amp;, x\in\mathbb R\setminus \{0\}\\ 0 &amp;, x=\infty  \end{cases}$ ? it appears to be a bijection and it's even simpler. If $F\supset E$ and $E\supset k$ are normal extensions. I want a counter-example where $F\supset k$ is not normal