Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$?
Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$ ?
I've been trying to sketch a proof by induction, but it seems more complicated that it should be.
Induction start is easy by considering $1 \times 1$, but in general there are many cases ?
Def: An elementary matrix is an $n \times n$-matrix corresponding to an elementary row operation.
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$\begingroup$There isn't, because they don't (for $n>1$).
Every invertible matrix is a product of elementary matrices. If invertible matrices commuted, then any two invertible matrices would commute!
Can you find an example of two elementary matrices which don't commute?
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