find nonsingular matrices $B$ and $C$ that satisfy $BC+ CB= 0$

How to find two non singular matrices $B$ and $C$ such that $BC+ CB= 0$. Clearly $B$ and $C$ must be square matrices of same order and the order is even.so it is possible we can find some example of order 2. But I cannot find any. I would be happy if someone helps me in finding some matrices.Thanks in advance.

Solution 1:

Hint: start from
$$ B=\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}. $$ To make $BC$ and $CB$ be of opposite sign, you need to change the order of columns in one $B$ and the order of rows in the other one. Can you think of some suitable permutation matrix $C$ for that?

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