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I have been stuck of this problem forever if any one can help me out it would be much appreciated. I need to express the given matrix as a product of elementary matrices.

$$ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 2 & 2 & 4 \end{pmatrix} $$

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    $\begingroup$ The title of a question should always indicate what type of question it is. Everybody posting questions here is seeking help, and is stuck, so your title doesn't give anybody any idea if they can help you or not. $\endgroup$ – Thomas Andrews Mar 23 '15 at 21:23
  • $\begingroup$ I have taken the liberty of changing the question title. $\endgroup$ – Mark Fischler Mar 23 '15 at 21:29
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    $\begingroup$ Have you tried inverting this matrix? If you keep track of your elementary row operations, it'll give you a clear way to write it as a product of elementary matrices. $\endgroup$ – Cameron Williams Mar 23 '15 at 21:29
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    $\begingroup$ You can tranform this matrix into it's row echelon form. Each row-operations corresponds to a left multiplication of an elementary matrix. $\endgroup$ – abcdef Mar 23 '15 at 21:31
  • $\begingroup$ Thank you very much Mark Fischler I appreciate it $\endgroup$ – zach Mar 23 '15 at 21:44
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To do this sort of problem, consider the steps you would be taking for row eleimination to get to the identity matrix. Each of these steps involves left multiplication by an elementary matrix, and those elementary matrices are easy to invert. Thus:

$$ \left( \begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ -2&0&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&1 \\ 0&2&0 \\ 2&2&4 \end{array} \right) = \left( \begin{array}{ccc} 1&0&1 \\ 0&2&0 \\ 0&2&2 \end{array} \right) \\ \left( \begin{array}{ccc} 1&0&0 \\ 0&1/2&0 \\ 0&0&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&1 \\ 0&2&0 \\ 0&2&2 \end{array} \right) = \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&2&2 \end{array} \right) \\ \left( \begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&-2&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&2&2 \end{array} \right) = \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&0&2 \end{array} \right) \\ \left( \begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1/2 \end{array} \right) \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&0&2 \end{array} \right) = \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&0&1 \end{array} \right) \\ \left( \begin{array}{ccc} 1&0&-1 \\ 0&1&0 \\ 0&0&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&0&1 \end{array} \right) = \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&0&1 \end{array} \right) \\ $$ So $$ \left( \begin{array}{ccc} 1&0&1 \\ 0&2&0 \\ 2&2&4 \end{array} \right) = \left( \begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 2&0&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&0 \\ 0&2&0 \\ 0&0&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&2&1 \end{array} \right) \left( \begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&2 \end{array} \right) \left( \begin{array}{ccc} 1&0&1 \\ 0&1&0 \\ 0&0&1 \end{array} \right) $$

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