I have $A\in \mathbb{R}^{3\times 2}, \ B\in \mathbb{R}^{2\times 3}\;$ such that $AB=\begin{bmatrix} 8 &amp; 2 &amp; -2\\ 2 &amp; 5 &amp; 4\\ -2 &amp; 4 &amp; 5 \end{bmatrix}$ How can I show that $BA=\begin{bmatrix} 9 &amp; 0\\ 0 &amp; 9 \end{bmatrix}$? I tried by bruteforcing systems of linear equations, but it seems too complex.
Let $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2\times3}(\mathbb{R})$ be matrices satisfying   $AB =\begin{bmatrix} 8 &amp;2 &amp;−2\\ 2 &amp;5 &amp;4\\ −2 &amp;4&amp; 5 \end{bmatrix}$. Calculate $BA$. (Golan, The Linear Algebra a Beginning Graduate Student Ought to Know, Exercise 426.)   Maybe it can be solved by solving a system of equations, but I think there is a shorter way since this problem was in my exam.  Thanks.
How do I technobabble without sounding like technobabble? On occasion, I've had reason to describe obviously fictional technical processes in-character. I'd like to be able to describe on these occasions without sounding like I'm saying utter gibberish. But it's not straightforward to do, particularly if I have to make stuff up on the spot.   How do I do it? Macro for figure position? Why is the following minimal example not working?  I.e., what do I need to do, in order to define the float placement specifier of a figure within a macro? (I know that I should use \newcommand, but in the "full" example the error is part of an involved "superfigure" command. \newcommand is not working too btw.)  \documentclass{scrreprt} \usepackage{blindtext}  \def\PosNotWorking{h} \def\FramboxNotWorking{5cm} \def\CaptionWorking{Caption is working.}  \begin{document}  \Blindtext[1]  \begin{figure}[\PosNotWorking]   \centering     \framebox[5cm]{My special figure. (Pos: \PosNotWorking)}             \caption{\CaptionWorking} \end{figure}  \Blindtext[1]  \end{document}   If I compile this, the figure is at the top (default behaviour). If I enter \figure[h] instead of \figure[\PosNotWorking], the figure is of course between the two paragraphs. How do I edit the boot order in the GRUB menu to have Windows first? I have a dual boot on my computer of Windows 7 and Xubuntu 12.10. The Linux boot is 1st and the Windows 2nd. I would like to reverse the boot order in the GRUB menu. I need to know the steps to go about editing GRUB. I do most of my work on Windows and need it to be 1st so I can boot up and not watch it. An insurance company supposes that the number of accidents that each of its policyholders will have this year is Poisson distributed, with a mean depending on the policyholder: the Poisson mean Λ of a randomly chosen person has a Gamma distribution with the Γ(2, 1)-density function fΛ(λ) = λe−λ (λ > 0). Find the expected value of Λ for a policyholder having x accidents this year (x = 0, 1, 2, . . .).  I am not sure how to incorporate the gamma parameters. So far I have found P(X = n|Λ = λ) = Poisson(λ) and fΛ(λ)= Exp(λ) so by integrating P(X = n|Λ = λ)fΛ(λ) then letting the integral of Gamma (n + 2, 2) equate to 1, I have worked out P(X = n) = (n + 1)/2^(n+2). How do you find the expected value of Λ from that? I have seen that a Gamma-Poison mix has E(X) = αβ which would give 2 but I have tried integrating fΛ(λ) and can't seem to get 2. I was thinking of the noted 1980 paper by Sidney Coleman and Frank de Luccia--"Gravitational effects of and on vacuum decay"-- about metastable vacuum states that could tunnel to a lower energy "true vacuum" with catastrophic results. I suspect the answer is that if our vacuum is a false one, we wouldn't "see" the true one til it hit us. I am looking to build a camera trap for wildlife viewing, able to capture pictures (or even videos) in decent quality to be able to identify distinct features. I do however need to be able to see the pictures in real time, so the camera would need some sort of wireless connection to a receiving device such as a phone.  I thought of using an FPV setup as used in drone racing, but the quality and range seems to be quite low, and the strengths of an analog signal seem to be latency and fps, which are both not important to me. It would be nice to have some sort of digital signal to be able to do some editing, such as storing pictures. It would also be cool to connect several cameras to one device.  Are there any solutions to this problem? Would a raspberry pi/arduino setup work in some way? Thanks! How could I get the terminal to be launched once I have logged in, so that typing commands is much simpler? Can an html element have multiple ids? is looking bad right now. No errors in the console, but the following files have no content:         The CSS on another subdomain works fine though: Why use a ReentrantLock if one can use synchronized(this)? For an infinite line charge, $E$ falls off with $1/r$; for an infinite sheet of charge it's independent of r! The infinitesimal contributions to $E$ fall off with $1/r^2$, so why doesn't the total $E$ fall off the same way for the infinite line and sheet charges? Why and how can you hear explosions in space in Star Wars? Considering that sound can't travel through vacuum. tkz-graph: changing individual vertex type I'm using these settings for most vertices:  \GraphInit[vstyle=Classic] \tikzset{VertexStyle/.append style = {minimum size = 5pt, inner sep = 0pt}}   But I'd like to have one or two vertices of my graph be hollow with a thick outer edge. I'm very confused about what options can be used in \Vertex, which ones can go in \SetUpVertex, for which I have to use \tikzset, etc. What's the best option here? I have decided to switch from Powerpoint to Beamer, and for my first presentation I'd like to use the , and the same colors as in this goggle slide template:  .  I tried to do some research about it, the closest answers I found are the question  and the . I tried this code:  \documentclass{beamer} \usepackage[frenchb]{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc}  \usetheme{metropolis}  \definecolor{bluegreen}{rgb}{3, 166, 155} \definecolor{pitchblack}{rgb}{0, 0, 0} \definecolor{lightbeige}{rgb}{255, 251, 241} \definecolor{mediumgray}{rgb}{183, 183, 183}  \setbeamercolor{background canvas}{bg=pitchblack} \setbeamercolor{normal text}{fg=lightbeige} \setbeamercolor{headline}{bg=bluegreen, fg=mediumgray}  \begin{document} \begin{frame}     \frametitle{The title}     Lorem ipsum... \end{frame} \end{document}   I does everything I want, except that the title bar's background color is lightbeige:    What did I do wrong? Or what argument should I pass to \setbeamercolor to get it right? Thanks in advance :-). Determining likely relationship based upon autosomal DNA shared cM Given two tests that share some cM of autosomal DNA, how can I easily find the predicted relationships that such shared cM would suggest? For example, a 2,200 cM match suggests it could be either: grandparent, half-sibling, or grandchild. Typically, to box in some text, I use the following:  \fbox {     \parbox{\linewidth}{     This is some text! Blah blah blah...     } }   However, say I have a verbatim block within that. It will never compile. Anyone know how I could draw a box around text which contains a verbatim block? I should note that I want to be able to draw a box around not just a verbatim block. For example, I would want to draw a box around something like:  This is text. \begin{verbatim} This is some more text. \end{verbatim} And this is even more text. After Windows 10 Fall Creators Update, a svchost.exe instance is constantly using CPU time Since installing the Windows 10 Fall Creators Update, I have a svchost.exe process that is constantly using about 14​ % of CPU time:    This svchost.exe instance is responsible for hosting three services:    The CPU time is consumed by a thread inside that process that  cannot attribute to any of these three services:    Given that these three services are considered critical system services, it's hard to do anything about them. These services cannot be stopped or disabled by services.msc, and terminating the process triggers a bugcheck (CRITICAL_PROCESS_DIED). Disabling the Windows Defender Firewall through the settings UI does not reduce CPU usage.  I'm out of ideas. What could be causing this? How can I debug this further? Is there a workaround to stop these services?    Update: After some debugging with , we've isolated this to the Windows Defender Firewall service. Process Monitor shows that it's constantly accessing the registry:    A CSV export of a Process Monitor capture, filtered to that process, is available .  I’m still looking for a way to stop that behavior.    Update 2: Analysis with the Windows Performance Analyzer shows that most CPU time is consumed by code from rpcrt4.dll: SHM with acceleration at mean position Suppose we have an equation of  motion as $$\frac{d^2x}{dt^2} = -kx + c,$$ then can it be called a SHM? Since acceleration is still proportional to displacement. But then, how will we define the mean position? I have the following problem: two groups A and B.  The proportions of married and single are ($p_1$) and ($p_2$) respectively, and the standard deviations ($s_1$) and ($s_2$). The sample consists of  ($n_1$) A and  ($n_2$) B observations.  Now I want to test whether the proportions are equal. I use the following formula to compute the z-statistic:  z = $abs(p_1 - p_2) / \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$  and test the hypothesis: h0: $p_1 \neq p_2$  An alternative approach I tried is to estimate the t-value based on the standard deviations (hence just comparing means), that is by the formula (assuming unequal variances):  t = $abs(p_1-p_2)/\sqrt{ s_1^2/n_1 + s_2^2/n_2 }$  My question are:    do you agree with my approach or; should it be two-sided or one-sided?   Thank you. UIActionSheet is not showing separator on the last item on iOS 7 GM