Involutory matrix $2 \times 2$ I want to find out how many  $2 \times 2$ involutory matrices are there over $\mathbb Z_{26}$. $ $ Is there any formula to calculate this?   $ $  Thanks for your help.
Why are there $736$ matrices $M\in \mathcal M_2(\mathbb{Z}_{26})$ for which it holds that $M=M^{-1}$? I'm currently trying to introduce myself to cryptography.  I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for encryption and the inverse matrix $M^{-1}$ for decryption. The book recommends to use a matrix $M$ for which it holds that $M=M^{-1}$, since we then have the same key for encryption and decryption.  In the book, they use a $2 \times 2$ matrix $M$ over $\mathbb{Z}_{26}$ as example, and state that for there are 736 $2 \times 2$ matrices for which it hold that $M=M^{-1}$.  I'm trying to pick up on as much as possible when reading things, since I find it counter-productive for learning to skip something, when you don't get the theory behind it. Can someone enlighten to me, as to why it is that there are 736 possible $2 \times 2$ $M=M^{-1}$ matrices and how to find them?
Elegantly determine if more than one boolean is "true" I am trying to add a list into an another nested list before adding values of a list to nested list it is getting the correct value but after adding i am getting (already output) in debug log when i checked.    public List&lt;String&gt; fieldName {get;set;}  public List&lt;List&lt;String&gt;&gt; fileValuesXML; // To get the field values of xml  for(Dom.XMLNode child : SalesforceObjecet.getChildElements()) {                        objecetName = child.getText();             system.debug('Object Name in XML field  ' + objecetName );              for(Dom.XMLNode subchild : child.getChildElements())              {                 system.debug('Sub child ' + subchild.getText() );                 for(Dom.XMLNode subchildfield : subchild.getChildElements())                 {                     system.debug('Sub child Field ' + subchildfield );                     fName  = subchildfield.getText();                     fieldName.add(fName);                     //fileValuesXML.add(fieldName);                     //system.debug('Filed Name in XML field  ' + fieldName);                 }                  system.debug('Filed Name before affing to filevalueXML  ' + fieldName);                                     fileValuesXML.add(fieldName);                 fieldName.clear();                 system.debug('Filed Name after affing to filevalueXML  ' + fileValuesXML);             }    }    I am getting values in debug log as:     Filed Name before affing to filevalueXML  (AccountSource, field3)   Filed Name after affing to filevalueXML  ((), (already output),   (already output)) It is well established that:   'In quantum mechanics, the behavior of an electron in an atom is described by an orbital, which is a probability distribution rather than an orbit.   There are also many graphs describing this fact: : (hydrogen atomic orbital - one electron) In the figure, the shading indicates the relative probability to "find" the electron, having the energy corresponding to the given quantum numbers, at that point.  My question is: How do these symmetries shown in the above article occur?                 What about the 'preferable' axis of symmetries? Why these? Kworker, what is it and why is it hogging so much CPU? Ubuntu 15.10 is not in Boot Manager I have a problem with Ubuntu 15.10  I have in my computer windows 10. Some days ago i decided to install Ubuntu in partition so i can have both in the same hard drive. The problem is that i cant get into ubuntu, i dont have the grub up so i always go to Windows in default mode. The only way to go into ubuntu is with the CD of installation, when im on the grub of the cd i press Esc, then i text exit in command lines and i go to a boot manager in which i can choose the ubuntu system. Its a strange think because it is the only way i can get into ubuntu, i will like to solve this problem. I try many tutorials already of solving this problem. I didnt get any solution so i will like to ask u before doing another tutorial. Thank u How do I find the caller of a method using stacktrace or reflection? dist$(x,A)=0$ if and only if $x\in \overline{A}$ (closure of $A$) Let $(X,d)$ be  a metric space and non-empty $A\subset X$. The distance from $x$ to $A$ is defined as $$\mbox{dist}(x,A)=\inf\{d(x,a):a\in A\}$$  I want to show that dist$(x,A)=0$ if and only if $x$ is in the closure of $A$.     (=>): dist$(x,A)=0$ implies $\inf \{d(x,a):a\in A\}=0$. Suppose $x\not\in \overline{A}$, then $x\in X\setminus \overline{A}$, which is an open set. Thus $\exists\epsilon &gt;0$ such that punctured open ball $B(x;\epsilon)\setminus\{x\}= \{y\in X:0&lt;d(x,y)&lt;\epsilon\} \subset X\setminus\overline{A}$. But then dist$(x,A)=\inf\{d(x,a):a\in A\}\ge \inf\{d(x,y):y\in B(x,\epsilon)\setminus\{x\}\}&gt;0$. This is a contradiction! Hence, $x\in \overline{A}$.      (&lt;=): Suppose $x\in \overline{A}$. If $x\in A^\circ$ (the interior of $A$) then $\inf\{d(x,a):a\in A\}=d(x,x)=0$.   If $x\not\in A^\circ$ then $\exists $ a sequence $(a_k)_k\subset A$ such that $a_k$ converges to $x$. Then, $\forall \epsilon &gt; 0, \exists N\in\mathbb{N}$ such that $k\ge N\implies d(a_k,x)&lt;\epsilon$.      So that dist$(x,A)=\inf\{d(x,a_k)\}=\lim\limits_{k\to\infty} d(x,a_n) = 0$.   Please let me know if my proof is rigorous / correct. I'm not completely sure, think it's better to check. I tried to search for similar questions, as I always do, but was not able to find one. I just created a new  on Data.Stack Exchange with my existing Google OpenID and it seems to be not associated with my regular Stack Exchange account. Is this by design? and if so, why? 2.5 GB usable out of 4 GB of ram Order of general- and special linear groups over finite fields. Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have?   $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$   Here GL is the , the group of invertible n×n matrices, and SL is the , the group of n×n matrices with determinant 1. How do I set read/write permissions my hard drives? MapThread with non-rectangular lists Variable number of arguments in C++? Really stuck with this, \begin{cases}  2tx'(t)-1=x^2t \\  x(t=1)=1 \\  \end{cases} I don't understand the variables. What does it mean that $x$ differentiated have $t$ as a function value, like this: $x'(t)$? Don't get it :( Should we pass a shared_ptr by reference or by value? Since  and  you can get into a situation where your max HP is zero due to leveling up… what happens next?  What happens when your max HP is 0 because you leveled up with a negative Constitution modifier? Are you dead, unconscious, or what, and where do the rules say this? (The  assumes death, but that isn't established as fact anywhere yet.) Declare variable to hold job name and if job running kill it I’m looking for a single file that does these things in order.   Declare a variable to hold a job name  Run a query to check whether that job declared is currently running Set up an if loop If the job is running, execute the stored procedure that kills it.   My question is to write a query that declare a variable to hold a job name and gives us result the job is still running or not if the job is running execute the stored proc to kill it.  My manager was asking me to listing running jobs and suspended jobs and declare a variable to hold job name and setup an IF loop like if job is running run proc to kill it. How to convert milliseconds into human readable form? Can I use dispensers or droppers to auto-replant crops? Script at Startup