TITLE: Does the set of 2x2 complex matrices form a field
QUESTION [5 upvotes]: Let $C$ be the set of all the matrices of the form $$ C = \{\begin{pmatrix}
z & -w \\ w 
 & z
\end{pmatrix} \; | \; z,\ w \in \mathbb{C}\}. $$
My question would be if the $C$ forms a field with an addition ($+$) and matrix multiplication ($\times$)? If not, why not?
I went through all of the field axioms and couldn´t find the issue but it doesn´t seem right.

REPLY [11 votes]: Well, that includes singular matrices like {{1, -i},{i, 1}}, so it won't satisfy the requirement that every non-zero element have a multiplicative inverse.