# Simplify a Determinant Identity

## Background

Suppose that $A$ and $B$ are singular and nonsingular matrices respectively. Simplify $\det((A+B)^2−(A−B)^2)$.

## Problem

A wrong solution with a vote of -2 is chosen by Daniel. Why can this happen?

## Possible explanation

That’s because he’s correctly done the expansion until $\det(2AB + 2BA)$.

## Raison d’être of this post

Having spent time on typing a comment, I worry that it will automatically disappear in sooner or later if the accepted answer is deleted. Therefore, I back it up here.

Consider $% $ % However, if $A = 0$ and $B = I_3$, then the answer is clearly zero. As a result, we can’t decude further from $\det(2(AB + BA))$.

## Lessons learnt

The generation of a random matrix/array of integers using randi([imin, imax], m, n). For more details, you may read GNU Octave’s manual.

# A Group of 689 Elements (2)

## Background

Two years ago, I thought about a group of 689 elements.1 I only managed to show the existence of such a group.

## Problem

Inspired by the use of Sylow III to show that a group of order 15 has only one structure: $\Z_{15} \cong \Z_3 \times \Z_5$, I wondered if $\Z_{689}$ is the only possible structure for a group of order 689.

# Indices in Definition of External Semidirect Product

## Problem

In my notes, the external semidirect product $G_1 \rtimes_\gamma G_2$ of two groups $G_1$ and $G_2$ with respect to a homomorphism $\gamma: G_2 \to \Aut G_1$, is defined as

\begin{multline} \forall\, x_1,y_1 \in G_1, \forall\, x_2,y_2 \in G_2, (x_1,x_2) \times_{G_1 \rtimes_\gamma G_2} (y_1,y_2) \\ = (x_1 \times_{G_1} \gamma(x_2)(y_1), x_2 \times_{G_2} y_2). \end{multline}

Why don’t we write $(x_1,y_1)$ and $(x_2,y_2)$ instead?

# A Quick Markdown Syntax Error Detection for Writing MathJax Equations in Octopress Posts (4)

## Background

I’m recently enhancing the $\rm \LaTeX$ code for inline limits. For the reason of doing so, you may refer to the external link of my recent linklog Inline Limit Rendering.

## Problem

In the previous post in this series written over one year ago, I have included a code block which enables deferred MathJax loading. However, I manually added this chuck of code in the HTML file generated by kramdown, which created the problem described in the next subsection A problem with Vim’s folding arised.

## Solution

Firstly, save the code for loading MathJax in the previous post in this series in a separate file ~/script.html. Then use the following commands within Vim in order to avoid leaving the current buffer and to improve efficiency.

## Problem

Similar to the previous post.

Owner@Owner-PC MINGW64 /c/github/blog2 (gh-pages)
\$ jekyll serve
WARN: Unresolved specs during Gem::Specification.reset:
pygments.rb (~> 0.6.0)
jekyll-watch (~> 1.1)
WARN: Clearing out unresolved specs.
Please report a bug if this causes problems.
C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/resolver.rb:35
7:in resolve': Could not find gem 'jekyll (~> 3.1) x64-mingw32' in the gems ava
ilable on this machine. (Bundler::GemNotFound)
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/r
esolver.rb:164:in start'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/r
esolver.rb:129:in resolve'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/d
efinition.rb:193:in resolve'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/d
efinition.rb:132:in specs'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/d
efinition.rb:177:in specs_for'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/d
efinition.rb:166:in requested_specs'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/e
nvironment.rb:18:in requested_specs'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler/r
untime.rb:13:in setup'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/bundler-1.7.2/lib/bundler.r
b:121:in setup'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/jekyll-2.5.3/lib/jekyll/plu
gin_manager.rb:37:in require_from_bundler'
from C:/Ruby200-x64/lib/ruby/gems/2.0.0/gems/jekyll-2.5.3/bin/jekyll:16:
in <top (required)>'
from C:/Ruby200-x64/bin/jekyll:23:in load'
from C:/Ruby200-x64/bin/jekyll:23:in <main>'


# Website Preview in Mobile Devices

## Goal

To know the page layout of a website, notably my blogs, on mobile devices before publishing it.

## Problem

I used ifconfig to check the IP address of my desktop where the preview site was hosted. It’s 192.168.1.5. When I typed in this address followed by a colon and the port number 4000, then the browser said “connection timeout” after loading for a while.

# My Host File Edited

## Background

A preview of a blog is often needed before it’s published.

## Problem

However, it is possible that one doesn’t like to type localhost in the address bar, and would like to have other names.

## Solution

The idea is in Local Setup – edit your hosts file in How to test localhost from any device on your network written by Wes Bos. On *nix, the file path is still the same as on OSX, but many users would rather use Vim for editing /etc/hosts. One may even use sed with the -i flag and the sudo privilege in order to directly manipulate this file.

• sudo sed -i "3i127.0.0.1\tblogtest.com" /etc/hosts for insertion of “127.0.0.1        blogtest.com” before the third line.
• sudo sed -i "3c127.0.0.1\tblogtest.com" /etc/hosts for changing the third line to “127.0.0.1        blogtest.com”.

Note that the tab is escaped as \t in these two commands.