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Let $A$ and $B$ be $4\times4$ matrices and suppose the following sequence of elementary row operations transforms $A$ to $B$: first swap row $1$ and $2$; second replace row $4$ by row $3$ plus row $4$; third multiply row $3$ by $-2$.

We can transform $B$ to $A$ by performing a certain sequence of elementary row operations based on the procedure above.

What is the matrix that realizes the FIRST of these operations?

A

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&1&1\end{pmatrix}$

B

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&-1&1\end{pmatrix}$

C

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-\cfrac{1}{2}&0\\\ 0&0&0&1\end{pmatrix}$

D

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-2&0\\\ 0&0&0&1\end{pmatrix}$

E

$\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}$

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