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.