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There I have found from an old tutorial notes from a Linear Algebra course I am taking that I am finding the question rather cumbersome. I haven't come across any elementary matrices problem set in this way before. I have tried to using Gauss-Jordan Elimination to reduce matrix A, whilst performing the row operations on the corresponding elementary matrix. Although, I am unsure if this is the correct approach as the problem notes that RREF calculation is not necessary.

If $[A|b]$ denotes the augmented matrix which corresponds to the system of linear equations \begin{align*} w−y &=−2\\ −w +x +z &=5\\ 2x − y +4z &= 1,\\ \end{align*} determine the 3 × 3 matrix E such that $[ EA | Eb ]$ is the reduced row echelon form of $[A|b]$. Note: You are asked to calculate the matrix E but you are not asked to calculate the reduced row echelon form.

Any help on how to approach this question or determine the matrix E would be highly appreciated.

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Seems to me that the most straightforward method is to row-reduce $[A\mid I_3]$ to obtain $[EA\mid E]$. Strictly speaking, by doing so you haven’t computed the RREF of $[A\mid b]$. Another possibility is to verify that $A$ has full rank, in which case $E$ can be computed by inverting the square matrix obtained by deleting the last column of $A$.

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