If $B$ has eigenvalues $1, 2, 3$ and $C$ has eigenvalues $4, 5, 6$, and $D$ has eigenvalues $1, 8, 9$, what are the eigenvalues of the $6 x 6$ matrix $$A = \begin{bmatrix} B &amp; C \\ 0 &amp; D \end{bmatrix}$$ $'0'$ means zero $3 x 3$ matrix So far I am only able to figure out that the trace of matrix $A$ is $30$ by adding eigen values of $B$ and $C$. Stuck after that can't use the determinant property as we don't know the other values of matrix $A$.
It is known that all $B$, $C$ and $D$ are $3 \times 3$ matrices. And the eigenvalues of $B$ are $1, 2, 3$; $C$ are $4, 5, 6$; and $D$ are $7, 8, 9$. What are the eigenvalues of the $6 \times 6$ matrix  $$\begin{pmatrix} B &amp; C\\0 &amp; D \end{pmatrix}$$ where $0$ is the $3 \times 3$ matrix whose entries are all $0$.  From my intuition, I think the eigenvalues of the new $6 \times 6$ matrix are the eigenvalues of $B$ and $D$. But how can I show that?
How to convert wstring into string? This question was recently split into different questions for each workaround. See   meta posts for more information.   Can you use NBT tags in commands? I’ve tried commands that normally work in Java Edition like: give @s minecraft:planks 30 1 {display:{Lore:[&quot;Wooden planks&quot;]}} give @s stone_pickaxe{Unbreakable:1b} give @s item_frame{EntityTag:{Invisible:1b}} summon villager ~ ~ ~ {Offers:{Recipes:[{buy:{id:&quot;emerald&quot;,Count:5b},sell:{id:&quot;diamond_sword&quot;}}]}} summon tnt ~ ~ ~ {Fuse:20s} Find out which application listens to specific key bindings? I've got a custom loop that I'm using to display some Real Estate listings that will be available within 60 days. 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I'm closing the loop and adding the pagination using the following code:   &lt;?php endwhile; // End the loop. Whew. ?&gt; &lt;?php wp_pagenavi(); ?&gt; &lt;?php wp_reset_query(); ?&gt;   How can I go about fixing this? May I summarize all answers on StackOverflow, I asked a . There are many good answers but I can not accept just one of it. I would prefer to create my own answer where I summarize all answers. Is that allowed? When speaking precisely or technically, one would say that "Homo erectus and homo sapiens are two species of hominid" rather than "Homo erectus and homo sapiens are two species of hominids." The hominid here should be singular because we are speaking about instances of a single class ("class" being used here in its broader sense, not in the sense of taxonomic grouping).  Now let's consider more common parallel constructions replacing the word species with other words.      Tide and Wisk are two brands of detergent.       Tide and Wisk are two brands of detergents.      Cats and dogs are two types of pet you can buy at Pets-R-Us.       Cats and dogs are two types of pets you can buy at Pets-R-Us.   I feel that using the singular noun to specify the class is more grammatically correct, but sounds stilted in conversation. Almost everyone I hear making these constructions uses the plural form. What are your thoughts about singular vs. plural here?  Edit  Here's a more extreme example that may help clarify the issue. Consider the following sentence:     It was my first attempt at calming an angry crowd, and I just        stood there while people were hurling all kinds of insults at me.   Here I think the singular insult sounds strange, if not downright wrong.  Further Edit  I just noticed in A Treasury for Word Lovers (Morton S. Freeman, 1983) a section entitled "Types of Errors." This is a book about (American) English usage by an English professor and editor, which purports to be a "practical guide for serious writers and readers." I wish I could say this discovery satisfies the question, but in fact I now feel somewhat farther from the truth, if there is any single "truth" in this matter. But at least I don't have to worry about being wrong when using the plural form of the class in these constructions. number of subsets of even and odd Elantech Touchpad not working on Ubuntu 16.04 and Arch Linux So I have and Asus R558UF notebook with an Elantech Touchpad. After installing Ubuntu 16.04, the touchpad worked once but after restarting, it failed to work and has not been working ever since.  This is output of xinput:   ↳ Elan Touchpad                            id=12   [slave  pointer  (2)]   In Mouse and Touchpad settings, the touchpad is enabled. What should I do to fix this? How long does mana from Burning-Tree Emissary stay around? Why is compactness so important? I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? Especially as stating "for every" open cover makes compactness a concept that must be very difficult thing to prove in general - what makes it worth the effort?  If it helps answering, I am about to enter my third year of my undergraduate degree, and came to wonder this upon preliminary reading of introductory topology, where I first found the definition of compactness. I have two user profile fields "Last Name" and "First Name". I also have a user list views. I want to expose "Name" as filter which can search for both Last Name and First Name. How can I do single filter of combination of those two fields? Can I create it from views UI?  One possible way is to create another profile field "Name" which will be hidden in the form. On user save, I would combine the two field values into the "Name" field and then expose it as filter in the views. But this solution is hard coding and it needs to write hooks. How to find which package provides a file? For example, I can't install package qcad because of error: "can't find qdict.h". How can I find the package which provides the file qdict.h so I can install the package qcad? How to change keyboard layout in Lubuntu 15.10? I have recently upgraded my PC from Lubuntu 14.10 to 15.10. Before the upgrade my keyboard layout matched my Danish keyboard. But after the upgrade I probably have a standard English/US layout.   Preferences/Language Support, doesn't let me configure the keyboard layout. Preferences/Keyboard and Mouse, only let me configure stroking delay and similar.    I don't know about iBus and fcitx, as far as the tooltip infomation tells me, its for more complex languages such as Chinese.  I don't have a US icon in the taskbar, no keyboard and/or language icon at all.  Things I have tried:   Running the following in a terminal works, but only until the next reboot:  setxkbmap -layout dk  I got the following parameter in the file /etc/default/keyboard:  XKBLAYOUT="dk"  Installing and running the app Lxkeymap changes the keyboard to Danish when I run it, but rebooting will change the layout back to US.   I don't want anything fancy, I just want to set my keyboard layout to Danish. How can I do that? I'm new to LaTeX and trying to understand the reason why I'm getting the warning:  `h' float specifier changed to `ht'   I'm not attempting to change the float property of anything - the warning is showing up if I attempt to simply encapsulate some words in quotation marks:  ``highly distributed''   As far as I'm aware, there's no open sections anywhere and the file seems to render properly. So realistically it doesn't matter if the issue gets solved or not; but I'd like to know the reason I'm getting the warning.  Thanks in advance. Date & time format in Krusader - is it changeable at all? Using Ubuntu 16.04. I like Krusader, but can't figure out how it is possible to change date/time format for the Modified column. I truied to change system regional settings, but Krusader ignores them (while, e.g., Midnight Commander uses system settings). Switching application language (in Help menu) also does not affect the date/time format. I like the 24-hour format, not PM/AM, and I may prefer even to hide time, and leave only date. Is that possible in Krusader?  The other post is of no help, as there is no even such file in mys systems as ~/.kde/env/setlocale.sh Is there any significant difference between Nikon and Canon? Why do researchers need universities? This is a follow-up question to  and probably an even more naive question. If I'm understanding the answer correct, professors need funding to do research, but once they get it, the university takes some of the grant as overheads (to pay for office space, electricity, etc), and the cut the university takes is substantial.  Given that then, why do professors need universities? One could just apply for the grant as per normal, and once one gets it, buy a slightly bigger house and convert one of the rooms into a lab. One loses nothing to overheads, gets to work from home, has zero teaching duties, can choose to settle anywhere (no two-body problem!), and can even monitor an experiment 24/7. This gets even easier if one works in a field that doesn't need a physical lab. Further, presumably the grant covers postdoc salaries, so one would still be able to pay for postdocs (although probably not PhD students since a professor without a university will not be able to award a degree).   The obvious answer is that one cannot apply for a grant without a university, but Google indicates that's not the case, e.g.  only looks at one's previous accomplishments &amp; potential applications of the research result, both of which are independent of the university. I suppose one could lose journal access, but there's always stuff like arXiv / ResearchGate, emailing the authors of the desired paper, or even Scihub (oops). It's conceivable that not working for a university loses one some prestige, since one can no longer claim to be a professor. However even then I'd expect at least some academics to choose this path, valuing the convenience &amp; extra research funding over prestige.  If the answer to this varies from field to field, I'm most interested in the sciences. Is it physically possible to build a long zoom (17-300 mm for example) at an aperture value of around 1.4 or 1.2? I'm curious if the physics allow a company to build such a lens of course with good IQ, minimal color aberration, maximum sharpness, and little to none lens vignetting or edge softness I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number and wondered how normal this was. The following is what I've done in my investigation so far.  In the first figure below, I have plotted the probability that some number below $10^k$ has $m$ indistinct prime factors.  The second figure shows the evolution of the probability of finding a number with $m$ indistinct prime factors as the numbers one is looking at grows, i.e. it is just another way to visualize the same data. I have left out all points of zero probability.  If one looks at the first part of the figure, it seems that the peak of the curve for some $k$ shifts toward higher $m$ as $k$ grows.   My first question is this: Does this behavior (i.e. the peak shifting toward higher $m$ as $k$ grows) continue indefinitely, or will the peak stabilize at some point? The latter seems unlikely but I can't be sure.  My second question is this: Approximately how fast does the peak-value $m_{peak}$ move as $k$ grows?  I briefly looked at the , but this involves distinct primes and I don't really know much number theory, so any help would be greatly appreciated! Canonical vs. non-canonical terminal input I am studying for an exam and I am confused as to how canonical vs. non-canonical input/output works in Unix (e.g., curses).  I understand that there is a buffer to which "line disciplines" are applied for canonical input.  Does this mean that the buffer is bypassed for non-canonical input, or does it simply mean that no line disciplines are applied?  How does this process differ for input and output operations?  In the curses programs I have worked with that demonstrate canonical input, the input typed by a user is automatically entered either after a certain number of characters have been typed or a certain amount of time has passed.  Are either of these things considered "line disciplines" or is this something else entirely?