Order of the centralizer of matrix

Question

Given matrices in $\Bbb{Z}_p$ equivalent to this form $$ \begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix} $$

What is the centralizer and its order?

Answer 1

Hint: A matrix $M$ will be an element of the centralizer if and only if it commutes with the matrix $$ \pmatrix{0&1\\-1&0} $$ since your set of matrices consists of those having the form $$ a \pmatrix{1&0\\0&1} + b \pmatrix{0&1\\-1&0} $$

---

Note that $$ \pmatrix{0&1\\-1&0}\pmatrix{m_{11} & m_{12}\\m_{21} & m_{22}} = \pmatrix{m_{21} & m_{22}\\-m_{11} & -m_{12}}\\ \pmatrix{m_{11} & m_{12}\\m_{21} & m_{22}}\pmatrix{0&1\\-1&0} = \pmatrix{-m_{12} & m_{11}\\-m_{22} & m_{21}} $$ These matrices are equal if and only if $m_{11} = m_{22}$ and $m_{12} = -m_{21}$.