# Order of the centralizer of matrix [closed]

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?

## closed as off-topic by José Carlos Santos, Mohammad Riazi-Kermani, Arnaud Mortier, David Hill, Matthew ConroyFeb 9 '18 at 22:46

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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 $m_{12} = -m_{21}$.
• Once you have figured out which elements commute with $\pmatrix{0&1\\-1&0}$, count them. – Omnomnomnom Feb 9 '18 at 15:41
• Therefore this form $$\begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}$$ is obtain? – User432477438 Feb 9 '18 at 15:56