I came a cross a problem I could not solve today:     If matrices $A$ and $B$ satisfy $A^4 = 0$ and $B=I-A$, prove that $$B^{-1}=I+A+A^2+A^3$$   I doubt they expect me to start calculating 4 matrices that would have a,b,c,d  values in them etc  I'd love to know how to work with a powered matrix :)  thanks in advance
Let $A$ be a squared matrix, and suppose there exists an $n\in \Bbb N$ in a way that $A^n=0$. Show that $I-A$ is invertible and that $(I-A)^{-1}=I+A+\cdots+A^{n-1}$    I don't have a clue where to start from.
Reduce spacing between subequations Can I "fill" an empty object? If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition  $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$  then is $f$ constant function? Wacom Tablet - Illustrator brush tool I currently got the intuos art wacom tablet so still learning how to use it. I'm currently in illustrator trying to draw out a map and the brush tool is doing something weird and I cannot figure out why. You can see in my image that where there are curves the black slightly spills over and is filling in half the curve with black. Brush setting is set up as a new calligraphic brush and the size is set to pressure, 3pt-3pt. Are bicategories and lax 2-categories the same? I am reading an older physics book that my professor gave me. It is going over Coulomb's law and Gauss' theorem. However, the book gives both equations with an $r^3$, not $r^2$, in the denominator. Can somebody please explain why it is given as r^3? An image is attached for reference.  Also for equation 1-24, can somebody please explain how the middle side is equal to the right side with the del operator? How can I fit my table of contents into a single page? When I use \tableofcontents I get all my contents in a single page except for a line that goes on the other page, how can I make all in the same page? I am studying different point estimate methods and read that when using MAP vs ML estimates, when we use a "uniform prior", the estimates are identical. Can somebody explain what a "uniform" prior is and give some (simple) examples of when the MAP and ML estimators would be the same? How to increase difficulty in Dungeon World I've been GMing for a couple of sessions now and I'm still struggling at making encounters or situations more difficult. I will describe dangerous situations and encounters in fiction but the mechanics make it too easy for my players to succeed.  For instance no matter how big I describe the Troll, if the fighter of the party charges him and swing his sword, roll+2 for Hack and slash, he'll easily hit the troll. Almost 40% chance to hit without consequences. What is "umask" and how does it work? I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? How can we read and write values in a multidimensional array in verilog, i had read in  regarding different operations that can be done on multidimensional array. like if there is a part of code like this  input [15:0] me;  reg [15:0] p_array [7:0]; reg abc_pqr [2:0];  abc_pqr[0] &lt;= me[0]; abc_pqr[1] &lt;= me[1]; abc_pqr[2] &lt;= me[1]; p_array[abc_pqr[0]] &lt;= me[0]; p_array[abc_pqr[1]] &lt;= me[1]; p_array[abc_pqr[2]] &lt;= me[2];   so is this code will work and will not produce any error?  What is the difference between these two while assigning value  "&lt;=" and "=", i had seen "&lt;=" in most of the places during passing values. Noticing that the "Human Verification" image did not show, I found that the src, , shows that sstatic.net is giving an SSL vertificate for *.stackexchange.com.. How Deep Are Bender's Insides? Bender often pulls large objects from his torso cupboard, but how deep or how much empty space is he capable of storing? Can he bend space around the perimeters of the frame of his torso's door? If that is the case, does it also bend time? I suppose he might bend in more ways than one... Vieta Jumping: Related to IMO problem 6, 1988: If $ab + 1$ divides $a^2 + b^2$ then $ab + 1$ cannot be a perfect square. The famous IMO problem 6 states that if $a,b$ are positive integers, such that $ab + 1$ divides $a^2 + b^2$, then $\frac{ a^2 + b^2}{ab + 1 }$ is a perfect square, namely, $gcd(a,b)^2$.  How about a modification of this problem:    If $a,b$ are (strictly) positive integers, such that $ab + 1$ divides $a^2 + b^2$, then $ab + 1$ cannot be a perfect square.    I am looking for a proof of the claim above, or a counter-example.  One possible approach towards a proof is the following:   Suppose there is such a pair $(a,b)$ as above, then by the famous IMO problem 6 from 1988, $\frac{ a^2 + b^2}{ab + 1 } = g^2$ where $g = gcd(a,b)$. Since $ab + 1$ is a perfect square, $a^2 + b^2 = c^2$ for some integer $c$. So that $(a,b,c)$ is a Pythagorean tripple, therefore there exists positive integers $n,m,l$ with $n$ coprime to $m$, such that $a = l(n^2 - m^2)$ and $b = 2lnm$ - by Euclids formula.  Then by plugging in $a$ and $b$ in terms of $n,m,l$ into the original equation, and solving for $l$, it is possible to obtain the following:  There exists positive coprime integers $n,m$ such that   $\frac{(n^2 + m^2 + 1)(n^2 + m^2 - 1)}{2mn(n+m)(n-m)}$ is a perfect square.  If we put the quotient above into a program, as in this python code snipet:  N = 1000  for n in range(1,N):     for m in range(n+1, N):         A = (n*n + m*m + 1)*(n*n + m*m - 1)         B = 2*m*n*(n+m)*(m-n)         if A % B == 0:             print(A/B)   The quotient always is 2, regardless of whether or not $n$ and $m$ are co prime! So if the very strong implication   $2mn(n+m)(n-m) \mid (n^2 + m^2 + 1)(n^2 + m^2 - 1) \implies \frac{(n^2 + m^2 + 1)(n^2 + m^2 - 1)}{2mn(n+m)(n-m)} = 2$    holds, then the original problem will be solved. How to influence the position of float environments like figure and table in LaTeX?   This is a general question and should collect useful answers for all users. I hope we can use this as a reference Why can Arwen decide her mortality? Prove $[0, 1]$ and $(0, 1]$ are equinumerous by use of bijection What my thought was that the bijection was a piecewise function:  $f(x) = 1/n+1$ if $x = 1/n$ for some $n \in \Bbb N$ and $x=1/n$ if $x \neq1/n$ for some $n \in \Bbb N$.  However, the textbook says this is incorrect. I don't actually see why. Was it a matter of brackets or the open variable $(0,1]$, which is what I tend to think. Can NDSolve handle discontinuous data? What is the difference between old style and new style classes in Python? Is it possible to add attributes at runtime or to change the value of an attribute at runtime?