Question

Is R = {[a a+b
a+b b ] |a,b belongs to Z} a subring of M2(Z)? Why or why not?

Answer 1

$\begin{pmatrix} 1 & 3 \\ 3 & 2 \end{pmatrix}\in R$ and $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}\in R$, but $\begin{pmatrix} 1 & 3 \\ 3 & 2 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}=\begin{pmatrix} 7 & 5 \\ 7 & 8 \end{pmatrix}\not\in R$. So $R$ is not a subring of $M_2(\mathbb Z)$.