# Determining an Elementary 3x3 Matrix E from an Augmented Matrix of a system of Linear Systems

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.

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 $$A$$.