# Row swapping through matrix multiplication

Let's say I have a matrix

\begin{bmatrix}a&b\\c&d\end{bmatrix}

What would I have the multiply the matrix above by to obtain the following?

**\begin{bmatrix}c&d\\a&b\end{bmatrix}

• Hmm, I see you may just have meant to multiply the given matrix by another matrix, not use Gauss Jordan operations. Is that what you meant? [If so I'll delete my irrelevant answer below...] – coffeemath Feb 5 '16 at 0:50
• yes, that is what I meant. – Yunae Feb 5 '16 at 3:22

We want $$\begin{bmatrix}A&B\\C&D\end{bmatrix}\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}c&d\\a&b\end{bmatrix}$$ This means $c=Aa+Bc$, $d=Ab+Bd$, $a=Ca+Dc$, and $b=bC+Dd$. Suggestion: $$\begin{bmatrix}A&B\\C&D\end{bmatrix}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$$
$$\begin{bmatrix}1&1\\1&1\end{bmatrix}\\ \begin{bmatrix}1&0\\0&1\end{bmatrix}\\ \begin{bmatrix}1&0\\1&0\end{bmatrix}\\ \begin{bmatrix}0&1\\1&0\end{bmatrix}\\ \begin{bmatrix}0&0\\1&1\end{bmatrix}$$