find the characteristic and minimal polynomial of the matrix  $$B=\begin{pmatrix} a &amp; 1 &amp; 0 &amp; 0 \\ b &amp; 0 &amp; 1 &amp; 0 \\ c &amp; 0 &amp; 0 &amp; 1 \\ d &amp; 0 &amp; 0 &amp; 0 \end{pmatrix}$$  where $a,b,c,d$ are real numbers   I calculated the characteristic polynomial: $x^4 - ax^3 - bx^2 - cx - d$. I don't know how to go about finding the minimal polynomial   thanks for help
of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.  It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are both equal to $f$. This can be seen quite easily by induction on the degree of $f$.  Does anyone know a different proof of this fact? I would love to see a graph theoretic proof or a non inductive algebraic proof, but I would be happy with anything that makes it seem like more than a coincidence!
What is the size of an enum in C? I'm creating a set of enum values, but I need each enum value to be 64 bits wide. If I recall correctly, an enum is generally the same size as an int; but I thought I read somewhere that (at least in GCC) the compiler can make the enum any width they need to be to hold their values. So, is it possible to have an enum that is 64 bits wide? Please don't throw this one out as a duplicate of  I'm not asking about the plurality of the noun immediately following those two words - I'm interested in exactly why the example below is problematic (context: I have several children, one named John)...     1: One of my children's name is John.   I can't see any obvious reason why singular name there is completely unacceptable, but it certainly doesn't sit well with me. It seems to turn on whether one of my children can be treated collectively as the "subject" of the Saxon genitive 's, but I don't see a problem with...     2: It's actually one of my children's, but you can sleep in this bed.   I realise that in practice we'd normally pluralise names in the first example. But of course, this implies that one of attaches to my children's names, which isn't really the same construction.  So - is there some kind of "rule, principle" debarring #1 above? Or is it just "one of those things"?    (Apologies if my later switch from brothers to children invalidates any comments or answers.) I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why   $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$   isn't a counter-example. Any ideas? Thanks! How does long-distance space travel work in the MCU? In Guardians of the Galaxy vol. 2, we see long-distance space travel. It involves piloting your ship into some sort of "jump point" that opens up in space, taking the shape of a honeycomb-like structure with a single hex missing.  The way it's describe, these points are fixed in space: early in the movie, the Guardians have to out-run an enemy fleet to reach the one they need; later on, we see them fly through a very long series of them in sequence. It's also made clear that doing too many of these "jumps" in a row has consequences for the passengers.  The process seems to be similar to, say, the Mass Effect relay setup, mixed with a bit of Star Wars hyperspace travel.  Is this the first time we have seen long-distance space travel in the MCU? Is there anything from the source material that explains how this works from a logistical standpoint? If there's any information on the "science" behind it, that would also be helpful. Proof of "continuity from above" and "continuity from below" from the axioms of probability One of the consequences of the axioms of probability ($\sigma$ field and probability axiom) is the "infinite subset" and "infinite union" property, I can't figure out how it follows from them.  if $A_1 \subseteq A_2 \subseteq \cdots$ then   $$P\left(\bigcup_{i=1}^{\infty}A_i \right) = \lim_{k \to \infty} P(A_k)  $$  If $A_1 \supseteq A_2 \supseteq \cdots$ then   $$P\left(\bigcap_{i=1}^{\infty}A_i \right) = \lim_{k \to \infty} P(A_k)  $$ Past participle form of "exit"? Are there any simple rules for choosing the definite vs. indefinite (vs. none) article? Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$ Is it possible that my laptop shut down because my mobile is too close? If I put my phone on my notebook's left corner, it shuts down. Can anybody give me reliable explanation as to why this is happening?? When do the differences between APS-C and full frame sensors matter, and why? Initial value problem $t\frac{dx}{dt}=x+\sqrt{t^2+x^2}$ Sum of nilpotent and element is a unit in ring? I can't tell where I went wrong, am still a beginner.    Let N be a nilpotent in a commutive ring , let X be any element.   I'll will be showing X + N is a unit .   Assume Y(X+N) =1   Then YX(N^n-1) =  N^n-1   Multiplying both side by  X+N   We get X(N^n-1) =  X(N^n-1)  Thus nilpotent summed with any element gives unit. The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$ Include ALL pdf graphics in document I have several plots saved as .pdf in a folder and I would like to know if there is a way of adding all of them in a document at once, without typing the path to each file. An example of a sequence of Riemann integrable functions $(f_n)$ that converges pointwise to a function $f$ that is not Riemann integrable. I have been given the example $$ f_n(x)= \begin{cases} 1, &amp; \text{if}\; x\in\{(x_k)_{k=1}^{n}\}, \\ 0, &amp; \text{otherwise.} \end{cases} $$ Here the sequence $(x_k)_{k=1}^{\infty}$ is the sequence that "enumerates" elements of $\mathbb{Q}$. The sequence of functions converges pointwise to $$ f(x)= \begin{cases} 1, &amp; x\in\mathbb{Q}, \\ 0, &amp; x\notin\mathbb{Q}, \end{cases} $$ which isn't Riemann integrable. However, I am not quite convinced that $f_n$ is integrable. I would like some help in seeing how $f_n$ is Riemann integrable. Also, I would like more examples of sequences of Riemann integrable functions $f_n$  that converge pointwise to a function $f$ that is not Riemann integrable. How can I check whether a variable is defined in JavaScript? How to check whether a JavaScript variable defined in cross-browser way?  I ran into this problem when writing some JavaScript utilizing FireBug logging. I wrote some code like below:  function profileRun(f) {     // f: functions to be profiled     console.profile(f.constructor);     f();     console.profileEnd(f.constructor); }   It works fine in FireFox/FireBug, but it reports error in IE8 RC1. So, I'd like to do some checking whether console variable exists in the execution environment.  Below code works fine in FireFox, but not in IE8 RC1.   function profileRun(f) {     if (console != undefined) {         console.profile(f.constructor);     }      f();      if (console != undefined) {         console.profileEnd(f.constructor);     } }   However, if I do it this way. It works in IE8 RC1. Why?   function profileRun(f) {     if (window.console != undefined) {         console.profile(f.constructor);     }      f();      if (window.console != undefined) {         console.profileEnd(f.constructor);     } }   Is there any cross-browser way to check it? Solve recurrence relation $ a_{n+1} = (n+1)a_n + 1 $ $a_0 = 1 \\ a_{n+1} = (n+1)a_n + 1 $  Could you help me solve this? And maybe someone know good source explaining how to solve recurrence relations? Why ; is called semicolon and what is its history? Semi- is a Latin prefix meaning "half" and colon is another punctuation. Should I therefore say that semicolon is a halved colon? Dropping z dimension programmatically It seems there is no geoalgorithm to drop z values in the QGIS processing toolbox. Can this be done programmatically using the Python console? At the moment I have to save the layer with unchecked 'Include z-dimension' checkbox (non automatic geometry type): What's the meaning of this present perfect sentence?