I want to prove that if $A,B\in \mathbb R^{n\times n}$ are s.t. $AB=I$ then also $BA=I$ (where $I$ is the identity matrix).  $$AB=I\implies BABA=BA\implies (BA)^2-BA=0\implies BA(BA-I)=0.$$  How can I conclude from here ? I know that $CD=0\not\Rightarrow C=0$ or $D=0$. So I can't continue. I also thought to do : $$BA(BA-I)=0\implies ABA(BA-I)=0\underset{AB=I}{\implies} A(BA-I)=0,$$ but it's not conclusive. Any idea ?  Edited  I can't use determinant.
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix.
Why am I asked for a credit card number to download a free app? I  want to download free app in my iPod touch device and i have the following doubts.  Kindly resolve me doubts.   when i try to download it asks my credit card security number. since i am trying to download free app , why do i need to provide my credit card security code? Will i be debited for downloading the free app from app store?   I cant able to find any information regarding this on apple.  Please provide me some references about this from apple and please resolve my doubts.  Thanks a lot. Is my sugar daddy scamming me Why some dragon claw doors do not open? Can you explain briefly the main concepts and command line tools used to manage file permissions? File: \boot\bcd Status: 0xc00000e9 windows 8 I am having this error for quite a few months. When the problem occurs, my windows freezes at boot time with black screen. After few reset attempts, sometimes it shows the entitled error. after few attempts, the windows starts but everything become slow. If I again restart, it works well. But at a point windows never starts and I have to reinstall it. I have already reinstalled 3/4 times in the past 2/3 months. I checked for any HDD errors with EASEUS Partition Master Home Edition with surface test, chkdisk at boot time etc for my C drive and F drive but no errors found. Startup repair also fails to solve the problem.  Windows Boot Manager   Windows failed to start. A recent hardware or software change might be the cause.   To fix the problem:  &nbsp;&nbsp;1. Insert your Windows installation disc and restart your computer.  &nbsp;&nbsp;2. Choose your language settings, and then click "Next." &nbsp;&nbsp;3. Click "repair your computer."  If you don't have this disc, contact your system administrator or computer manufacturer for assistance.   &nbsp;&nbsp;File: \Boot\BCD  &nbsp;&nbsp;Status: 0xc00000e9  My laptop's info:  Model: Dell n5110 corei5 OS: Windows 8 PRO 64 bit Anti-virus:  Avast Home edition  Additional info: My Program Files, Program Files (x86) and Users directory is located at F drive and linked with symbolic link (mklink). Also I have changed these basic locations from regedit. I also use eclipse android intelx86 emulator, virtualbox and Genymotion emulator most often. Protons and Neutrons Overshoot Actual Mass? When I add up the mass of 6 protons and 6 neutrons in amu, I get a mass that is greater than the mass of carbon. I thought that it should be the other way around, because I have not including binding energy when I add up the mass of the protons and neutrons.  Proton: 1.007276466812 u  Neutron: 1.00866491600 u  6(1.007276466812)+6(1.00866491600)=12.099  Carbon: 12 u  Why is this so? Root Android virtual device with Android 7.1.1 Bounded sequence with divergent Cesaro means Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)?  More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist?  I encountered this problem while studying the limit of average payoffs criterion for ranking payoff sequences in infinitely repeated games. Wrapping a mesh around point cloud with cavities Given two date ranges, what is the simplest or most efficient way to determine whether the two date ranges overlap?  As an example, suppose we have ranges denoted by DateTime variables StartDate1 to EndDate1 and StartDate2 to EndDate2. My understanding is that GPUs are more efficient for running neural nets, but someone recently suggested to me that GPUs are only needed for the training phase. Once trained, it's actually more efficient to run them on CPUs.  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I have come to understand that is not necessarily the best thing to do and so found out how to do everything I need using simple JavaScript. I also store a variable in the initial page that then gets used in the imported html.  Questions: By switching from:  &lt;?php include "_header.html";?&gt;   to  &lt;script type="text/javascript" src="_header.js"&gt;&lt;/script&gt;   I am noticing that the "view source" shows the header.js link, whereas the PHP method will show the source as if the html had been there all along. Will search engines recognize the html in the JavaScript just as they would when loaded with PHP?  I understand it is bad practice to alter the page contents -after- it is downloaded, but this doesn't fall into that category does it? That would detail things like after a wait function, or after user interaction, correct?  Are there any other foreseeable downfalls to switching to JavaScript and turning off the forced PHP processing? Health decreasing for no apparent reason I am playing Dragonborn DLC. I am currently going through one of the black books quests and even though I am not fighting my health bar keeps going down. I can heal back up to full again but it keeps disappearing. If I do not heal I will die in about 30 seconds.  I have no disease or nothing is visible anyway, no clue as to what's wrong. I can leave the black book quest or fast travel to a different location or rest for 24 hours but still the health disappears. A generalisation of the fundamental theorem of algebra We already know the following theorem by d'Alembert and Gauss, often called fundamental theorem of algebra.  Theorem Let $P$ be in $\mathbb C[X]$ of degree $1$ or greater. There exists $\alpha\in \mathbb C$ such that $P(\alpha)=0$.  Can we give the following generalisation for polynomials with several variables?  Generalisation Let $n$ be in $\mathbb N^*$ and $P$ be in $\mathbb C[X_1,\ldots,X_n]$ of degree $1$ or greater. There exists $\alpha\in \mathbb C^n$ such that $P(\alpha)=0$.  Any references, hints or solutions would be greatly appreciated. Lightning Community Does Not Support pubsub for Lightning Web Components? Has anyone successfully used the pubsub module from lwc-recipes to enable inter-component communication in a Lightning Community?  The registerListener function in the pubsub module throws an error because the pageRef property is undefined.  Or is pubsub designed for Lightning Experience only? Where to start with Oracle? How to use `ybar interval` without skipping the last value I have the following code  \documentclass[tikz]{standalone} \usepackage{pgfplots} \usepackage{pgfplotstable} \usepackage{filecontents}  \begin{filecontents}{testdata.dat}   -3.316988309367815e-07 0.07000000000000001   -1.948815911147526e-08 0.13   7.15864175355987e-08 0   1.799432483450127e-07 0   0.0002850606647384129 0   0.0009429559686380238 0.08999999999999997   0.00157188462064142 0   0.005259918700865051 0   0.01739392641273605 0.18   0.06208364670743995 0.55   68.69485148165926 6.970000000000001   31.217611224923 79.12   0 12.89 \end{filecontents}  \pgfplotsset{compat=newest} \pgfplotstableread{testdata.dat}\datatable  \begin{document} \begin{tikzpicture} \begin{axis}[   ybar interval,   width=10cm,   height=5cm,   enlargelimits=false,    xtick=data,   ymin=0, ymax=100,   xticklabels={0 -- 0.063,0.063 -- 0.125,0.125 -- 0.25,0.25 -- 0.5,0.5 -- 1,                1 -- 2,2 -- 4,4 -- 5.6,5.6 -- 8,                8 -- 11.2,11.2 -- 16,16 -- 22.4,22.4 -- 31.5},   x tick label style={rotate=60,anchor=east}   ]   \addplot table[x expr=\coordindex,y=0]{\datatable};   \addplot table[x expr=\coordindex,y=1]{\datatable};  \end{axis} \end{tikzpicture} \end{document}   and the output looks like this:    But the command ybar interval does skip the last value. It should look like:    I prefer to have the last value to show up in the interval 22.4 -- 31.5. How do I achieve this without inserting a row of zeros to my table? Center image element in parent div Solve $y'-\int_0^xy(t)dt=2$ I have not idea how to approach this differential equation.   $$y'-\int_0^xy(t)dt=2$$.  Basically, I did,  $$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$  I am stuck.  Thank You.