Given two square matrices $A, B$, when is $$\det(A+tB) = 0$$ for all $t\in \mathbb{R}$?  An easy sufficient condition is that $A$ and $B$'s kernels have nontrivial intersection. Per Henning's comment below, this is not also necessary.  Does there exist a nice necessary and sufficient characterization?
I'd like to know if the following problem of elementary linear algebra is already solved / solvable.  For two (singular) $n\times n$ matrices $P$ and $Q$, if $\det(\lambda P+\mu Q)=0$ for any $\lambda,\mu\in\mathbb{R}$, what are conditions on $P$ and $Q$?
Adding feature class to Map in CURRENT project using ArcPy in ArcGIS Pro? Given a set of vectors $v_1 \cdots v_n$, the $n\times n$ Gram matrix $G$ is defined as $G_{i,j}=v_i \cdot v_j$  Due to symmetry in the dot product, $G$ is Hermitian.  I'm trying to remember why $|G|=0$ iff the set of vectors are not linearly independent. How do I find the common ratio of a geometric sequence? A geometric sequence has its first term equal to $12$ and its fourth term equal to $-96$.   How do I find the common ratio? And find the sum of the first $14$ terms convexity proof of a function including ln and sums Merge around a face So, I'm working on a control station that has buttons. It is a one-mesh creation, that means, I don't want to create buttons and things like that separated from the main mesh, but as you can see, I have problems. I would like to make the highlighted parts to be one with the main part around them. I've tried everything, and I have this problem on multiple creations. Any idea?  Update: I've added another example of the problem. I want them to be one face, instead of two. (2nd picture) What is a reliable method to locate floating islands in Terraria? How can I make a part of a texture glow? How to see time stamps in bash history? Is there any way I can see at what time the commands were executed from the bash history? We can see the order but is there any way I can get the time also?  Bottom-Line: Execution time in the Bash history "Interested with the course" or "interested in the course"? Interested in/with the course   Which preposition is correct? in or with, or both. Under the  tag I have seen many questions on earth-616. Are there any other versions of earth as well? How do they differ from each other? Motley fool says you can make $15,978 more per year with Social Security. Is this for real? The Motley fool has an ad campaign that says you can make $15,978 more per year on Social Security by doing 17 minutes of work using a legal loophole.  I don't want to link to their ad (although, you can find it by searching ).  What are they talking about here?  Is there any truth to this?  If someone could explain it to me without me having to watch the entire ad video or subscribe to their newsletter, I'd appreciate it. Finding functions such that $f(g(x)) = f(x)g(x)$ Let's assume the functions in question are continuously differentiable. If $f(g(x)) = f(x)g(x)$, then: $$f'(g(x))g'(x) = f'(x)g(x)+f(x)g'(x)$$ I attempted to find a particular solution by letting $g(x) = x$. Then, $$f'(x) = xf'(x) + f(x)$$ $$df = xdf + fdx$$ $$\frac{df}{f} = \frac{dx}{1-x}$$ $$\ln(f(x)) = -\ln(1-x)+C$$ $$f(x) = \frac {A}{1-x}$$ Where $A$ is a constant. This solution was fairly unsatisfying to me, and upon checking it, I see it doesn't actually work, as letting $g(x) = x$ leads to $f(x) = xf(x)$. I was wondering if there were better answers, or at least a less trivial way to go about the problem. How can author deal with late reviews? I know of 2 basic kinds of default web icons for websites:   : place an .ico formatted file at the root of your site named favicon.ico and most desktop browsers will find it and display it in various contexts (bookmarks, tabs). Can also be added in code, which allows you to specify other names and other graphics types to the file, though there are :  &lt;link rel="shortcut icon" href="/favicon.ico" /&gt; &lt;link rel="icon" href="/favicon.ico" /&gt;  : place a png formatted file called either apple-touch-icon.png or apple-touch-icon-precomposed.png at the root of your site and iPhone OS/iOS versions of Safari will pick those up and store them when the site is added to the Home Screen. Can also be added with code like:  &lt;link rel="apple-touch-icon" href="/custom_icon.png" /&gt;    Are there any other default icon graphics for a website a webmaster can include? How to Start Web Mapping? I want to start working on a web map at work to map some of our statistics by city.    I have an idea, but I'm not exactly sure where to start.   Are there any definitive resources that provide step by step guidance on how to do this? Default class inheritance access According to my text book:  The Lorentz Force F on a charged particle moving in an electromagnetic field is given by:  $$F= qE + q(v\times B)$$  then it states that "it is to be pointed out that only the electric force does work, while the no work is done by the magnetic force which is simply a deflecting force."  How can this by true won't the the magnetic force have a considerable magnitude, will it now impart kinetic energy to the charge? Any explanations will help a lot. Please keep the terminology simple, thanks. Supersymmetry in Quantum Field Theory Why does no one notice the Joker's presence as a policeman during the mayor's speech? Does commutativity imply Associativity?