Talk:Elementary Row Operations as Matrix Multiplications

I believe that the redlink "Transform" may be to be something with a more specific name, as there are umpty-finity concepts in maths with the title "transform". In this context what you need to link to is the definition of Matrix Product or whatever it is. --prime mover 01:18, 26 February 2012 (EST)

Ideally, the result would also hold for right-multiplication. This would be very useful for Determinant of Matrix Product/Proof 1, as one can take simply left-multiplication for the one, and right-multiplication for the other. --Lord_Farin 03:42, 16 March 2012 (EDT)

Actually, I don't know if it does hold for right-multiplication; I honestly didn't think of that. I'll see what I can do. --HumblePi (talk) 12:36, 3 December 2016 (EST)

No, it would not hold. Take $\mathbf A = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right], \; \mathbf E = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right]. \; \mathbf E$ is an elementary matrix, then using the conventional matrix multiplication:

$\mathbf {AE} = \left[ \begin{array}{ccc} 1 & 2 & 1 \\ 1 & 2 & 1 \\ 1 & 2 & 1 \end{array}\right]$ which is an elementary column operation on A. --HumblePi (talk) 13:08, 3 December 2016 (EST)

Future challenge: prove that post-multiplying by a given elementary matrix produces the corresponding elementary column operation to the elementary row operation that you get by pre-multiplying by the same elementary matrix. --prime mover (talk) 18:24, 13 June 2020 (EDT)