Prove that that set of all quaternions, with the matrix addition an multiplication, is a ring with unity. Find an example to show that this ring in not commutative. You may assume matrix addition and multiplication are associative and obey the distributive law.
$\alpha = \left[ {\begin{array}{cc} a+bi & c+di \\ -c+di & a-bi \\ \end{array} } \right] $
So I need to prove that this is commutative with addition, isn't commutative with multiplication, and is associative and distributive under multiplication. It is given that addition and multiplication are associative and distributive, thus I only need to prove/disprove commutative.
Addition would be $\alpha_1+\alpha_2$ and $\alpha_2+\alpha_1$? And then multiplication would be something like taking $\alpha_1\cdot\alpha_2$ and then $\alpha_2\cdot\alpha_1$? Then I would find the multiplicative identity to prove unity.