How to "flip" and change the sign of one particular row of this matrix?

Question

I would like to transform the following matrix : $\mathbf A$ =$\ \begin{bmatrix} a&b\\ c&d\\ e&f\\ g&h \end{bmatrix}\ $ into this one : $\mathbf B$ = $\ \begin{bmatrix} g&-h\\ e&-f\\ c&-d\\ a&-b \end{bmatrix}\ $.

I can easily transform each column separately by doing these operations :

$ \left\{ \begin{array}{rcl} \begin{bmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{bmatrix} \ \begin{bmatrix} a\\ c\\ e\\ g \end{bmatrix} = \begin{bmatrix} g\\ e\\ c\\ a \end{bmatrix}\\ \begin{bmatrix} 0&0&0&-1\\ 0&0&-1&0\\ 0&-1&0&0\\ -1&0&0&0 \end{bmatrix} \ \begin{bmatrix} b\\ d\\ f\\ h \end{bmatrix} = \begin{bmatrix} -h\\ -f\\ -d\\ -b \end{bmatrix} \end{array} \right. $

But doing this wont provide with the matrix $\mathbf B$. Sure I can concatenate the two matrices after doing two separate operations. But I would like to do this in only one operation...

Does anyone know how to do this?

Thanks in advance.

Answer 1

Multiply on the right by $$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and on the left by $$\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{pmatrix}.$$