Let $A \in R^{n \times n}$  Show that if $A =\alpha I$ for some $\alpha \in R$, then for all $B \in R^{n \times n}$ we have $AB = BA$  I proved this by:  $$AB = \begin{bmatrix} \alpha &amp; \cdots &amp; 0 \\ \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; \cdots &amp; \alpha\end{bmatrix}\begin{bmatrix} b_{11} &amp; \cdots &amp; b_{1n} \\ \vdots &amp; \ddots &amp; \vdots \\ b_{n1} &amp; \cdots &amp; b_{nn}\end{bmatrix} = \begin{bmatrix} \alpha b_{11} &amp; \cdots &amp; \alpha b_{1n} \\ \vdots &amp; \ddots &amp; \vdots \\ \alpha b_{n1} &amp; \cdots &amp; \alpha b_{nn}\end{bmatrix}$$  $$BA = \begin{bmatrix} b_{11} &amp; \cdots &amp; b_{1n} \\ \vdots &amp; \ddots &amp; \vdots \\ b_{n1} &amp; \cdots &amp; b_{nn}\end{bmatrix}\begin{bmatrix} \alpha &amp; \cdots &amp; 0 \\ \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; \cdots &amp; \alpha\end{bmatrix} = \begin{bmatrix} \alpha b_{11} &amp; \cdots &amp; \alpha b_{1n} \\ \vdots &amp; \ddots &amp; \vdots \\ \alpha b_{n1} &amp; \cdots &amp; \alpha b_{nn}\end{bmatrix}$$  $$AB = BA$$  How do I prove the opposite case? Show that if $B \in R^{n \times n}$ we have $AB = BA$, then there exists a $\alpha \in R$ such that $A = \alpha I$?
How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?  Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices.
Enabling GPU rendering for Cycles? I want to enable GPU rendering, but there is no option in User Preferences > System:    Why is this? How can I get cycles to render using my GPU? Custom right-click action in Unity Launcher keeps cursor busy for 20 seconds To have a "minimize window" option through right click on an icon in the Unity launcher (Ubuntu 14.04), I followed the detailed instructions  about changing a .desktop file and made a custom Firefox launcher in ~/.local/share/applications/ folder. The relevant part of the .desktop file is:  Actions=NewWindow;NewPrivateWindow;Minimize  [Desktop Action NewWindow] Name=Open a New Window Exec=firefox -new-window OnlyShowIn=Unity;  [Desktop Action NewPrivateWindow] Name=Open a New Private Window Exec=firefox -private-window OnlyShowIn=Unity;  [Desktop Action Minimize] Name=Minimize Windows Exec=sh /home/myusername/Documents/xdotool_sh/minimize.sh firefox OnlyShowIn=Unity;   The desktop action "Minimize" invokes a simple shell script, minimize.sh that has the following content:  #/bin/bash name=$1 for i in $(xdotool search --class "$name"); do     xdotool windowminimize $i done   The script uses xdotool, that can be installed from the official repositories, to find all the firefox windows, iterate over them and minimize them.  The script works and the launcher right menu option "Minimize Windows" works as well, but as soon as the windows are minimized, the mouse pointer gets in "busy" mode and stays like this for about 20 seconds (although the mouse actions are still responsive).   Does anyone know why starting a shell script from a right menu option in Unity could lead to this behaviour?  EDIT: Apparently the wait period is inevitable, as explained in Jacob Vlijm's answer. Since the mouse remains responsive, avoiding the transformation of the pointer in the spinning wheel is a partial esthetic workaround, as explained on .  EDIT2: giving the system a fake window is a better solution, as explained by Jacob below. Why is my Wi-Fi connection speed much slower than a wired connection? When I plug my laptop via LAN cable and check speed on speedtest.net I have over 32 Mb/sec. When I unplug the cable and connect to the same router I have less than 15 Mb/sec. I tried to change channels already but this didn't help. I'm sitting just near the router.   What can it be? The router is a Dlink 400. How to close/open a view panel I can't find a way to close a view panel, in particular the horizontal at the bottom, is pretty useless to me at this moment. How should we cope with 12 month old waking in the night? We are trying to get our 12 month olds sleep routine improved now that both of us will be returning to work.  We've been working on getting the settling routine sorted out each night, and haven't quite yet got to him falling asleep on his own, but he does now settle quickly at the end of the routine in our arms.  We will be moving to trying to get him to settle in his cot soon.  Once settled he will tend to do about 3-4 hours in his cot before waking screaming.  When cuddled he will settle down in about 10 minutes and can generally be put back in his cot.  However often he will wake several times.  Last night he went down at 8, woke at 10, woke at 12, woke at 1am.  Each time putting him back in his cot.   At that point we gave up and brought him into our bed where he slept through to 6am.  When waking up he is often on either all fours or standing in the cot whilst screaming.  The main problem is that he's no longer sleeping "solidly" and will often poke and prod his mum in his sleep, and she therefore doesn't get a great nights sleep.  And with us both back to work we both need our sleep as much as possible. I really like the new sidebar. With the new top bar eliminating the &quot;Hot Questions&quot; area there, I think this is the perfect place for it. One-click access to time-wasting questions about topics I may or may not have an interest in. Seriously, it's a procrastinator's dream come true. The only suggestion I can add is to extend it. I assume the &quot;hot&quot; algorithm can spit out more than 12 questions, and since we now have a huge amount of room for it(without being restricted to a weird drop-down scrollbox), why not show more? If there's something else planned for that spot, disregard. At a minimum, the title could link to the , as ShaWizDowArd suggested in comments. I've been using SE for a couple years now, and I've never seen that page, so it would add visibility there.    Update: They added a link! Just noticed this, I'm pretty sure it wasn't there yesterday. When you click it, it loads 12 more. At the moment, the link disappears once clicked, so you can see a total of 24 here now. Not a link to the full list, but better: How can momentum but not energy be conserved in an inelastic collision? In inelastic collisions, kinetic energy changes, so the velocities of the objects also change.  So how is momentum conserved in inelastic collisions? How to get pixel data from a UIImage (Cocoa Touch) or CGImage (Core Graphics)? I have a question about the pattern "turn &lt;something&gt; on &lt;somebody&gt;" :     Then he allegedly turned the body-building equipment on his 44 year-old wife, identified by sources as Feng Liang, and his elderly mother-in-law, Feng Kun Zaehng,77, the sources said.     I understand the phrase "*turn the gun on &lt;somebody&gt;", am not sure about "turn the equipment on &lt;somebody&gt;".  Could the example be wrong? If Z is a square integrable random variable such that $E[Z]&gt;0$, then prove that  $$P(Z\geq k) \geq \frac{(E(Z)-k)^2}{E(Z^2)} \forall k \in [0,E(Z)]. $$  This is a homework question and I don't even understand why the inequality is reversed comparing to Markov/Chebychev inequality. Should Cauchy-Schwartz inequality be used? If yes, then how to begin? Thanks a ton! What is a good Photoshop book for photographers? I am interested in not too basic techniques (which I can more or less figure out by myself), which on the other hand give good results, without the need to be an expert pro. If you view the Earth from far enough away can you observe its past? How was Loki not crushed by Mjolnir? In the Thor movie, Thor puts Mjolnir  on top of Loki at the end of the movie to stop him from moving.  Why did this not crush Loki?  And if it did not crush him, why could he not just roll out from under it? Google Maps Location History is showing an incorrect location “He is small enough to walk.” But baby wasn't walking so why didn't they use “small” instead of “small enough”? Image with axis Are there any keyboard shortcuts for Windows 10 developer preview's virtual desktops ("Task View")? Will Windows 10's reset delete my Linux partitions? I want to use the "reset my PC" feature of Windows 10 to wipe the data on my Windows partition. But I have also a Linux system installed, which I want to keep.  Will Windows 10's full reset wipe all my disk, or just Windows's partitions ? Can command substitution be nested in variable substitution? Round a double to x significant figures