The following stumps me:     Let $\mathbb K$ be a field. Let $A, B \in \mathbb K^{n \times n}$ where $B$ is nilpotent and commutes with $A$, i.e., $A B = B A$. Show that $$ \det(A+B)=\det(A) $$   I have no idea how to approach this I thought perhaps raise both sides to a power but nothing works. Thanks for all help.
Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy  $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$  Prove that $$\det(A+B)=\det B.$$
What home printer would allow for exact alignment of front and back? Note: I'm not a graphic designer, but this seemed like the most relevant place to post. Feel free to send me off somewhere else if I'm not in the right place.  I would like to make some notebooks with double-sided grid paper that I've already designed, but my printer seems like it doesn't allow for absolutely exact alignment. Even if one side prints nice and straight, the other side will be slightly skewed left to right by one to two millimeters, which I read from an earlier post is a pretty reasonable variance for a home printer. So I suspect it's not possible to get both sides completely aligned on this printer – but if I'm wrong, please let me know how.  What kind of printer would allow for absolutely exact alignment? Is there any difference between inkjet and laser in this regard, or a more expensive printer versus a cheaper one? Although we didn't do any changes, this new error appeared in the GSC Structured Data section:    I don't really get what it means. There are many fields for Schema.org, is homeLocation suddenly a required field? Does that make any sense? Restore All Databases Script I am migrating SQL Server DBs to a new instance.  I have been told that it is possible to dynamically build a RESTORE script from available backups in the system catalog.  Does anyone know of a sample script to do this?  Thanks! I have a lot of Excel sheets for all sorts of things - but none has ever acted up like this one here:    This used to be a totally normal Excel 2007 sheet which I was able to resize, maximize etc. - but not anymore. The MDI child window is fixed, both in position and size. I cannot move it around, more can I maximize it to use the full size of the screen anymore. I cannot double-click on it's title bar to maximize it either :-(  What happened? What can I do to "undo" whatever I did to it.  Any ideas? I tried to share / undo sharing, I tried to protect / unprotect the sheet - nothing seems to work... How to use figure inside a minipage? I'm trying to include an image, and keep it centered, and prevent from wrapping from the next page.  The following code works for that:  \begin{minipage}{\textwidth}     \begin{center}         Caption for image          \includegraphics[scale=0.5]{myimage.png}      \end{center} \end{minipage}   I'd like this image to be in a figure, with a caption.  The following doesn't compile:  \begin{minipage}{\textwidth}     \begin{figure}         \includegraphics[scale=0.5]{myimage.png}         \caption{Caption for image}         \label{fig:sample_figure}     \end{figure} \end{minipage}   I get the errors:  ! LaTeX Error: Not in outer par mode. ! Undefined control sequence.   How can I fix this? I am currently in the process of learning how to prove statements by induction. For the most part, I understand but I am at most clueless when it comes to prove some inequalities.  For example, I am having a hard time proving that: $n &lt; 2^n$ is true. (Even if it's clear that it is indeed true!)  In the induction step, I am stuck with: $n + 1 &lt; 2^{n + 1}$  I don't know what would be the assumption I should make from this statement.  Anyone could point me in the right direction? Thanks! Possible Duplicate:        When I click 'Unanswered' in the menubar, I get a list of questions, all of which have been answered. Does it just mean there's no accepted answer? If so, can we make that more obvious? Can we get some way of finding really unanswered questions? Please note: I posted this question about an hour ago on my account (I just made it as I am new to the site). I revisited my post and saw that I had not given definitions of what I was asking (I thought they were common notations) and it would not let me edit the post for some reason so I will ask here now and hopefully someone can delete the other thread.  How to prove for a finite group $G$ that $|\langle g \rangle|=o(g)$?  Here are the definitions:  Let $G$ be a finite group and $g \in G$  $o(g)$ is the smallest $m \in \Bbb{N}$ such that $g^m=e$ (the identity element of the group). (If such an $m$ exists)  $\langle g \rangle=\{g^m : m \in \mathbb{Z}\}$  I really don't know how to show that this is true I tried to say let $o(g)=m$ and then show that $|\langle g \rangle|$ has exactly $m$ elements but I wasn't even sure it did because it would contain the elements $g^0,...g^{m-1}$ so that would mean it's true but what about $g^{-1}$ and the other elements with negative coefficients so it seems like it would actually be bigger than $m$?  Thanks. maximum subarray of an array with integers Workaround for Android 4.4 not being able to write to SD cards? Yesterday I upgraded my Samsung Galaxy S4 to Android 4.4 KitKat and had the bad surprise that it cannot write to my SD card anymore... I know this problem is widely documented everywhere, but has any workaround been found for it yet? (Other than removing the card physically from the phone and opening it on my PC) Also, because of the KNOX flag I can't downgrade to 4.3...  (PS: Rooted or not, it makes no difference) Using all the letters of the word ARRANGEMENT how many different words using all letters at a time can be made such that both A, both E, both R both N occur together . SQL Server Stored Procedure Update not working with ASP.Net C# GridView RowUpdating How can I find my Bug-report that I submitted from the Desktop on launchpad?  I have found a partial solution to my own question:  The program that sends the Bug-reports from the desktop is called apport. It doesn't requires a launchpad account. The service whoopsie submits the Bug-Report directly to the server daisy.ubuntu.com (I'm not sure if those bugs actually appear on launchpad publicly.)  But I found my Bug-Report on my Harddisk in /var/crash/ .  From there I can copy and paste parts of the file to my opened bug on launchpad. Thanks for your tips.  Original question:  I had a severe bug, and I found someone with the same problem in launchpad. After I found a temporary fix I was able to submit the bug with the ubuntu-bug-reporter.  How can I find this ubuntu-bug-reporter generated bug-report in launchpad? I would like to combine those two bug-reports (mark as duplicate). Because the generated one has all the debug information and the other one has the description of this bug. Can I add extension methods to an existing static class? I'm a fan of extension methods in C#, but haven't had any success adding an extension method to a static class, such as Console.  For example, if I want to add an extension to Console, called 'WriteBlueLine', so that I can go:  Console.WriteBlueLine("This text is blue");   I tried this by adding a local, public static method, with Console as a 'this' parameter... but no dice!      public static class Helpers {     public static void WriteBlueLine(this Console c, string text)     {         Console.ForegroundColor = ConsoleColor.Blue;         Console.WriteLine(text);         Console.ResetColor();     } }   This didn't add a 'WriteBlueLine' method to Console... am I doing it wrong? Or asking for the impossible? Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$. Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.  So $(a^2+b^2)(c^2+d^2) = a^2c^2+a^2d^2+b^2c^2+b^2d^2$  and $(ac+bd)^2 = a^2c^2+2acbd+b^2d^2$  So the problem is reduced to proving that $a^2d^2+b^2c^2\ge2acbd$ but I am not sure how to show that My wife wants a 4th baby, but I don't Find all roots of $\,x^2\!\equiv x\pmod{\!900}$, i.e all idempotents in $\Bbb Z_{900}$ I need the idempotent elements of $Z_{900}$ $2^2\cdot 3^2\cdot 5^2=900$ Of course there's $$0 \pmod 4 \\ 0 \pmod 9 \\ 0 \pmod {25} \\ $$ and $$ 1 \pmod 4 \\ 1 \pmod 9 \\ 1 \pmod {25} \\ $$ I found the answers by making a C++ program to test all numbers, but I don't know how to quick solve on paper. This is the output of the program $(0, 0, 0) \rightarrow 0 \\ (1, 1, 1) \rightarrow 1 \\ (0, 0, 1) \rightarrow 576 \\ (0, 1, 0) \rightarrow 100 \\ (1, 0, 0) \rightarrow 225 \\ (1, 1, 0) \rightarrow 325 \\ (0, 1, 1) \rightarrow 676 \\ (1, 0, 1) \rightarrow 801 \\$ Unable to preserve ownership in a copy but able with a move? I ran dolphin like this:  dolphin . &amp;   I see messages in the console like these:  QPixmap::scaled: Pixmap is a null pixmap   How can I keep the console clean or silent? $\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \mathrm dx$ Evaluate Integral Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \ dx$$