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

Question

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.

Answer 1

Let $B= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ and $C = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}$,

then $$BC=\begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$$

and $$CB=\begin{bmatrix}0 & 1 \\ - 1 & 0 \end{bmatrix}.$$

Hence, we have $BC+CB=0$

I have used Pauli matrices.

Answer 2

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?