5

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.

variableXYZ
  • 1,053
  • 6
  • 13
  • 3
    If you restrict to $z,w\in \mathbb{R}$ then this is a representation of $\mathbb{C}$ so then it's a field – Jojo Mar 15 '22 at 07:50

1 Answers1

11

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.

Tim Goodman
  • 293
  • 2
  • 9
  • 4
    It may also be of interest that if you replace one of the columns in your definition with its complex conjugate, you will get something where every non-zero element has a multiplicative inverse - in fact, you get a representation of the quaternions. But it's still not a field, because quaternion multiplication isn't commutative. – Tim Goodman Mar 14 '22 at 21:39