Let $S$ be the set of all $3\times 3$ matrices over $\Bbb Z$ such that the product $AA^t=I$. Then find $|S|$?  Taking $A=$\begin{bmatrix} a &amp; b &amp;c \\d &amp; e&amp; f\\g &amp; h &amp; i\end{bmatrix} we get $3$ relations $a^2+b^2+c^2=1;d^2+e^2+f^2=1;g^2+h^2+i^2=1;$  How should I count from here?
Find three real orthogonal matrices of order $3$ having all integer entries.   I have no idea to solve the problem. I don't know how to start. If $A$ be such matrix then $AA^T=A^TA=I_3$. Please help me.
I took a bunch of photos at my son's martial arts club using a D90 with a 50mm F1.8.  The club is lit with overhead fluorescent lighting, and I was getting some weird results where some shots are "white", others have an "orange cast", and still others are partly white and partly orange in the same shot.  I suspect it has to do with the combination of the frequency of the light flicker and with the camera shutter speed. Here is a sequence of 3 shots that show the problem.  The shots were taken with 1/500s at F2 and ISO-800 and with Auto-WB.  First shot ("white/normal"):    Second shot ("orange cast" at top):   Third shot ("orange cast" at bottom):   These were taken in burst mode within a second of each other.  Can anybody tell me what is going on?  And, how I can avoid this? Spacing changes when using unicode-math range feature. Why? You can use the range feature of unicode-math to set the math font for a specific unicode slots or the macro accessing the glyph. I tried this with the sum sign &amp; Latin Modern, see my example:  \documentclass{scrartcl}  \usepackage{amsmath} \usepackage{unicode-math} % use unicode in math &amp; setup math fonts \setmathfont{Asana Math} %\setmathfont[range=\sum]{Latin Modern Math} % substitute Sum sign  \begin{document}  Notice the different sum sign $\sum$, as intended.   Notice also, the difference in spacing, not intended: \begin{equation}     \begin{bmatrix}         0           &amp; \frac{1}{2}   &amp; \frac{2}{3} \\         \frac{3}{4} &amp; 0             &amp; \frac{4}{5} \\         \frac{5}{6} &amp; \frac{6}{7}   &amp; 0     \end{bmatrix}     \begin{pmatrix}         A \\         B \\         C     \end{pmatrix} \end{equation} \end{document}   However, not only the sum sign, but also the spacing in the matrices changes. With Latin Modern commented out (as in the example):  With Latin Modern active (remove % before relevant line):  Why is that? BTW, I'm using Lualatex from TeXLive 2012. HTTP and HTTPS impacts on SEO I have a website http://www.domain_1.com/ hosted in a shared webserver.  I have used SSL for another website https://www.domain_2.com/ on the same server.  Due to some technical limitations (single IP), if you try https://www.domain_1.com, browser will show the same domain_2 website. But this is not a concern for me.  Question: will this scnario impact SEO of domain_1 (non-HTTPS) site? How can I prevent system clipboard image data being pasted into a WPF RichTextBox The new profile shouldn't mistakenly imply that you can earn the Informed badge on a meta site How do I fix a "Problem with MergeList" or "status file could not be parsed" error when trying to do an update? Is “what’s” a correct short form of “what does”? E.g. “What’s he think?”  Usually ’s is short for “is” so I don’t know. What kinds of desktop environments and shells are available? Switch command output language from native language to english If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$ How to perform hard / cold reboot of iPhone or iPad? I understand that when I press and release the power button on the top of my iPhone or iPad, it stays on but turns off the screen.  Whereas if I press and hold the power button, a "slide to turn off" control appears on the screen.  Sliding this, as I was told, does turn the unit off, but it does so by suspending the RAM to disk (flash memory, actually.) So, when turning on the iPhone or iPad, it isn't so much rebooting the OS as it is resuming it from flash storage.  How to force the OS to completely reboot on these devices? Bring all planes in mesh to same level I have added a plane in top view. I have then extruded the plane according to a floorplan in topview.  I don't know what happened, but when I switched my view, I could see that some parts of it are for some reason high up in the air.  I would like to ask how I can bring everything back down onto the grid / floor.  And if somebody could tell me how this might have happened, I would also be glad. The Drupal 8 REST UI configuration page asks me to select Authentication providers for GET method; otherwise, it doesn't allow me to export the REST request.  How can I provide anonymous user to access my REST GET API without providing any auth information? If asking a question or giving instruction towards a group of objects where the number is unknown (and possibly could be one) what is the correct way to phrase the sentence? Which of the following sentences are more correct:     Has there been any studies regarding this and if so what has been the   conclusion?   OR     Has there been any studies regarding this and if so what have been the   conclusions? For some reasons my brush stroke suddenly looks like this:     As the picture showed, the edges are jagged and pixelated. It doesn't happen before and it's just one of the built-in inking brush.  I know there are a lot of similar questions here but please don't mark this as duplicate. I double checked all the thing the other answers suggested: It's brush tool, not pencil, and my mode of brush and layer is Normal, not Dissolve. It's pixelated even if I draw with mouse so it's not tablet driver issue. I disabled GPU render but nothing changed. How do I fix this? How to change the monitor's refresh rate? How do I change the monitor's refresh rate in Oneiric?   There is no option to do it in the Displays dialog now. By default it's 60Hz, but I need 75Hz on my monitor.  In previous releases there used to be such an option in the Displays dialog.  Edit - Half baked solution  The best I could do so far was:  xrandr -s 1280x1024 -r 75   This is perfect, but my problem is that it isn't permanent. It defaults back to 60 Hz after logging out and back in.  Perhaps this should go into some script and added to startup? I just don't know how to do that. What resolution should a large format artwork for print be? For wall-sized graphics and large banners (e.g. 3m × 5m), what is an acceptable PPI for print? As I understand it 300 ppi is the typical for 'small' artworks (esp. for clean text resolution). However, for small artworks, the audience usually engages at a much closer distance. Therefore, in the interests of keeping a bitmap file down to a reasonable size, can you get away with a much-reduced PPI for a large format/banner print? Android App initial release date When you are viewing an app in Google Play store, it displays the following : Updated, Size, Installs, Current Version, Requires Android, and Rating.  I am trying to do some research and figure out the initial release / publish dates of different apps. Is there a way to get this information from Google Play, or perhaps a website with this information? how to adapt the sorting of a custom type `@standard` in using biblatex/biber Seeking GeoServer tutorials? What are sources where I can learn how to use GeoServer?  I know of these two sites:    - which in fact is not working (at least at the moment)  - not only (tutorials) for GeoServer   But I am interested in more complex information, and not to read only documentation.