Can every elementary row op on a matrix be represented by multiplying that very matrix by an elementary matrix?
Elementary row operations versus matrices
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1Yes. Every elementary row operation can be performed by left multiplication by an elementary matrix. Just as every elementary column operation can be performed by right multiplication by an elementary matrix. – 2011-11-05
1 Answers
The answer to your question is yes. I will give you the matrices in each case: Let $M$ be an $n \times n$ matrix.
1) Multiplying row $i$ of $M$ by a non-zero scalar $c$ corresponds to multiplying $M$ on the left by the matrix $X$ where $X$ is the same as the identity matrix, but the $(i,i)^{th}$ entry replaced by $c$.
2) Interchanging rows $i$ and $j$ of $M$ corresponds to multiplying $M$ on the left by the matrix $X_1$ where $X_1$ is the identity matrix with the $i^{th}$ and $j^{th}$ diagonal entries replaced by $0$ and the $(i,j)^{th}$ and $(j,i)^{th}$ entries replaced by $1$.
3) Adding $c(row \hspace{1mm}j)$ to $row \hspace{1mm} i$ corresponds to multiplying $M$ on the left by the matrix $X_2$ where $X_2$ is the identity matrix with $(i,j)^{th}$ entry replaced by $c$.
Hope this helps. Column operations correspond to multiplying by elementary matrices on the right.
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1Note also that the elementary matrix corresponding to a given row operation can be obtained by doing that row operation to the identity matrix of the correct size, and likewise for column operations. – 2011-11-05