Answer to Question #203305 in Abstract Algebra for Dag stern

Question

Check if matrix

1 a b

[A ]= 0 1 c

0 0 1

Where a,b,c∈R

is an abelian group with respect to matrix multiplication.

Answer 1

Let us check whether the set $A=\Big\{ \begin{pmatrix}1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1\end{pmatrix}:a,b,c\in\mathbb R\Big\}$ is an Abelian group with respect to matrix multiplication. It is sufficiently to prove that $A$ is a subgroup of the general linear group $GL_3(\mathbb R).$

Let $\begin{pmatrix}1 & a_1 & b_1\\ 0 & 1 & c_1\\ 0 & 0 & 1\end{pmatrix}, \begin{pmatrix}1 & a_2 & b_2\\ 0 & 1 & c_2\\ 0 & 0 & 1\end{pmatrix}\in A.$

Taking into account that

$\begin{pmatrix}1 & a_1 & b_1\\ 0 & 1 & c_1\\ 0 & 0 & 1\end{pmatrix}\cdot \begin{pmatrix}1 & a_2 & b_2\\ 0 & 1 & c_2\\ 0 & 0 & 1\end{pmatrix}= \begin{pmatrix}1 & a_2+a_1 & b_2+a_1c_2+b_1\\ 0 & 1 & c_2+c_1\\ 0 & 0 & 1\end{pmatrix}\in A$ and

$\begin{pmatrix}1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1\end{pmatrix}^{-1}= \begin{pmatrix}1 & -a & ac-b\\ 0 & 1 & -c\\ 0 & 0 & 1\end{pmatrix}\in A,$ we conclude that $A$ is a subgroup of $GL_3(\mathbb R),$

that is $A$ is a group with respect to matrix multiplication.

Since $\begin{pmatrix}1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}\cdot \begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\end{pmatrix}= \begin{pmatrix}1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\end{pmatrix}$ and

$\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\end{pmatrix}\cdot \begin{pmatrix}1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}= \begin{pmatrix}1 & 1 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\end{pmatrix},$ we conclude that the group $A$ is not Abelian.