# Let A and B be A=[[1,0,-2],[3,1,0],[1,0,-3]], B=[[1,0,-3],[3,1,0],[1,0,-2]] Find an elementary matrix

Let A and B be $A=\begin{bmatrix}1&0&-2\\3&1&0\\1&0&-3\end{bmatrix},\ B=\begin{bmatrix}1&0&-3\\3&1&0\\1&0&-2\end{bmatrix}$
Find an elementary matrix E such that EA=B.

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The given matrices are $A=\begin{bmatrix}1&0&-2\\3&1&0\\1&0&-3\end{bmatrix},\ B=\begin{bmatrix}1&0&-3\\3&1&0\\1&0&-2\end{bmatrix}$
Note that the matrix A can be obtained by interchanging the rows $$\displaystyle{R}_{{1}}$$ and $$\displaystyle{R}_{{3}}$$ of B.
Perform the same operation in $I=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ and obtain that $R=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}$
Check the condition as follows.
$EA=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}\begin{bmatrix}1&0&-2\\3&1&0\\1&0&-3\end{bmatrix}$
$=\begin{bmatrix}1&0&-3\\3&1&0\\1&0&-2\end{bmatrix}$
$$\displaystyle={B}$$
Thus , the required elementary matrix is $E=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}$