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

Edmund Conti 2021-11-14 Answered
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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Expert Answer

Phisecome
Answered 2021-11-15 Author has 8010 answers

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 \[E=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}\]

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