# Elementary matrices

Is there a way to visualize the action of elementary matrices? (Or perhaps matrices in general). Perhaps someone could give an intuitive view of the effects of elementary row operations.

Actually I am trying to intuitively understand why these elementary matrices general $GL_n$.

Thanks.

-
The $i$-th column of a matrix $A$ shows you what $A$ will do to the $i$-th unit vector. Knowing this it should be easy to see what geometric operation your elementary matrices represent. – Listing Nov 5 '11 at 11:49

1. When you act on geometric figures with the elementary matrix with a single non-one entry on the diagonal, you are stretching/shrinking the figure in the direction of that axis. If the entry is negative a reflection is also included. If the absolute value of that entry is $>1$, it's a stretch, if $<1$, you're shrinking things.
2. The elementary matrix swapping two rows gives the $n$-dimensional analogue of swapping the $xy$-coordinates on the plane. IOW, you are reflecting w.r.t to the line $x=y$.