I don't recall this well from linear algebra so I just need some quick confirmation. If $A$ is an $n \times n$ matrix and B is also $n \times n$ and $AB=I_n$ does it follow that also $BA=I_n$? I think no but then what do we call inverse matrix: the left inverse, or the right one, or only when the left and right are the same we talk about inverse matrix of $A$? Also, do we define inverse matrices of non-square matrices?
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.
How to prevent Google from finding my admin index page? Why is James Bond less tech savvy in the Daniel Craig movies? $m$ is a perfect square iff $m$ has an odd number of divisors? How to create live autofill search? I am currently trying to create a wordpress search function that shows live results underneath the search bar. There is an example on the  website (screens below). I'm not looking for an autofill like you would find on Google.com which completes words you type, rather I want it to find actual posts on the site.   I tried doing a scrub through Wordpress Answers and other similar resources but have only run into implementing a Google type search which is not what I am looking for. Any help or points in the right direction would be greatly appreciated. I created some texture maps in MindTex to use in Blender, but I can't figure out how to properly hook them up. The maps I have:  - Diffuse  - Normal  - Height   - Specular   - Gloss   - Self Illumination  - Reflection  - Ambient Occlusion   MindTex (left) vs. Blender (right)  As you can clearly see it's a pretty big difference.  is my material node setup in Blender. In the examples I don't actually use self illumination or reflection images, but including this in the answer would be helpful. Using fontspec with Luatex it is possible to create a fake slant and use that as italic should no italics be around:  \setmainfont[ AutoFakeSlant=0.15 ]             {FontName}   However, also if an italic version exist this faked slant will be used both for \itshape and \slshape. How can I keep using the shipped italic font defined by the ItalicFont= parameter to \setmainfont but use fake slant for \slshape? I'm Swedish, but living in Switzerland at the moment. Can I open a US bank account? Either when in the US on vacation or through the internet/phone?  There are lots of situations when I find myself in need of an american credit card or just a bank account. When ordering things online (most retailers refuse to accept my orders if I try to pay with my Swedish or Swiss credit card, even if I ship it to an US address... ) etc etc. How can I create a new extensible symbol? Given an arbitrary thin vertical symbol, can I create a new math delimiter from it (and if so, how)?   Edit: Ideally, I would like to be able to create an extensible symbol, either built out an existing normal symbol, or even made from scratch.  Edit 2: I am pretty sure that what I am after is an explanation of how to create a .tfm file for an extensible symbol (which Werner's answer describes the general structure of, but not how to make one).    For example, let's say I wanted to use the \dagger () or \wr () symbol as a delimiter, and write expressions like    \[\left\ldagger \sum_{n=1}^\infty a_n \right\rdagger\]   where the symbol scaled appropriately. Would there be any distinction between declaring a symmetric delimiter (like \vert), as opposed to one that has inherently different "left" and "right" versions like [ and ]?  Now, even being unaware of the inner workings of TeX, I imagine that there has to be lots of information in the fonts for how a delimiter is to be scaled in different situations, and if I try to make my own delimiter, this information won't be present. Would I need to get deep into FontForge or equivalent to achieve my custom delimiters, or is there an easier way? How can I prevent iPhoto/Photos/Image Capture from launching everytime I plug in my iPad/iPhone? Normal blocks I can mine fine, but when it comes to diamond and gold ores, the block just gets "destroyed" (as in you can't pick it up). I'm playing on multiplayer with no spawning monsters. How to switch between Ubuntu and Windows 7 without a reboot I have installed windows 7 and then Ubuntu 11.04. Now I want to know how I can switch from Ubuntu to Windows 7 and from Windows 7 to Ubuntu without restarting the computer. Can I run CAT5/6 cables parallel to electrical cables? The derivative of $x!$ and its continuity is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain.  Our teacher said the factorial of a fraction is the fraction itself. He also said the graph is continuous but could not determine the derivative of $x!$. In the same way as you can currently render the active viewport in Open GL? Some aspects of the Workbench renderer look as if they will be very useful for NPR people. 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. Can you get a Windows (AD) username in PHP? I have a PHP web application on an intranet that can extract the IP and host name of the current user on that page, but I was wondering if there is a way to get/extract their Active Directory/Windows username as well. Is this possible? In  thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that  $$f(f(x)) = f(x) + x$$  (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is the  $\frac{1+\sqrt{5}}{2}$).    Having read about  shortly before though, I came up with this train of thought:  $$f(f(x)) = f(x) + x$$ $$\Leftrightarrow f^2 = f^1 + f^0$$ $$f^2 - f - f^0 = 0$$  where $f^n$ denotes the $n$'th iterate of $f$.  Now I solved the resulting quadratic equation much as I did with plain numbers  $$f = \frac{1}{2} \pm \sqrt{\frac{1}{4} + 1}$$ $$f = \frac{1 \pm \sqrt{1+4}}{2} = \frac{1 \pm \sqrt{5}}{2}\cdot f^0$$  And finally the solution $$f(x) = \frac{1 \pm \sqrt{5}}{2} x .$$  Now my question is: *Is it somehow allowed to work with functions in that way?** I know that in the above, there are denotational ambiguities as $1$ is actually treated as $f^0 = id$ ... But since the result is correct, there seems to be some correct thing in this approach.  So can I actually solve certain functional equations like this? And if true, how would the correct notation of the above be? Questions about proving $\lnot \lnot a = a$ This is all in the context of a Hilbert system with modus ponens $A, A\to B \vdash B$ and axioms:  Axiom $1$: $A \to (B \to A)$   Axiom $2$: $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$   Axiom $3$: $(\lnot B \to \lnot A) \to (A \to B)$  So I tried to prove double-negation elimination by doing the following:  \begin{array}{ccc} 1 &amp; \{ \lnot \lnot a \} \vdash \lnot \lnot a &amp; \text{Premise}\\ 2 &amp; \{ \lnot \lnot a \} \vdash \lnot \lnot a \to (\lnot \lnot \lnot \lnot a \to \lnot \lnot a)  &amp; \text{ Axiom } 1\\ 3 &amp; \{ \lnot \lnot a \} \vdash \lnot \lnot \lnot \lnot a \to \lnot \lnot a  &amp; \text{ Modus Ponens } 1, 2\\ 4 &amp; \{ \lnot \lnot a \} \vdash (\lnot \lnot \lnot \lnot a \to \lnot \lnot a) \to (\lnot a \to \lnot \lnot \lnot a)  &amp; \text{ Axiom } 3\\ 5 &amp; \{ \lnot \lnot a \} \vdash \lnot a \to \lnot \lnot \lnot a  &amp; \text{ Modus Ponens } 3, 4\\ 6 &amp; \{ \lnot \lnot a \} \vdash (\lnot a \to \lnot \lnot \lnot a) \to (\lnot \lnot a \to  a)  &amp; \text{ Axiom } 3\\ 7 &amp; \{ \lnot \lnot a \} \vdash \lnot \lnot a \to  a  &amp; \text{ Modus Ponens } 5, 6\\ 8 &amp; \{ \lnot \lnot a \} \vdash a  &amp; \text{ Modus Ponens } 1, 7\\ 9 &amp; \vdash \lnot \lnot a \to a  &amp; \text{ Deduction Theorem } \\ \end{array}  And then I continued on, to prove double negation introduction:  \begin{array}{ccc} 10 &amp; \vdash \lnot \lnot \lnot a \to \lnot a &amp; \text{Double Negation Elimination}\\ 11 &amp; \vdash (\lnot\lnot \lnot a \to \lnot a) \to (a \to \lnot \lnot a) &amp; \text{ Axiom } 3\\ 12 &amp; \vdash a \to \lnot \lnot a &amp; \text{ Modus Ponens } 10, 11\\ \end{array}  My questions:   Am I allowed to invoke the deduction theorem in the middle of a proof like this (step $9$)? Is this considered a context change? Is context-changing allowed? Are the proofs even right/correct? Assuming it's all correct: Are we able to combine them to say that $a \to \lnot \lnot a$ and $\lnot \lnot a \to a$, therefore $a \iff \lnot \lnot a$ or is this symbol not allowed? What ultimately let us say $\lnot \lnot a = a$, the use of the equal sign here? Is there a term denoting the writing of words using numbers instead of letters? Sponsor Gallery, Page, or Something Like That?