Exercise
8
[1ZZ]Consider the ring of matrixes \(ℝ^{2× 2}\) let’s define
\[ A={\begin{pmatrix} 0
& 1
\\ 1
& 0
\end{pmatrix} }\quad ,\quad B={\begin{pmatrix} 0
& 1
\\ 0
& 0
\end{pmatrix} }\quad , \]
then check that
\[ AB={\begin{pmatrix} 0
& 0
\\ 0
& 1
\end{pmatrix} } \quad ,\quad BA={\begin{pmatrix} 1
& 0
\\ 0
& 0
\end{pmatrix} }\quad ; \]
you conclude that the ring of matrixes is not commutative.