When you multiply a matrix $A$ on the left by an elementary matrix $E$ to obtain $EA$, it performs a row operation on the matrix $A$. We can view both matrices as parallelogram, where the we use the rows of matrix to create the parallelogram. *The rows of $A$ are the edges of the parallelogram in the image on the left. *The rows of $EA$ show up as the transformed parallelogram in the image on the right. This second image includes both the original matrix $A$ and the new matrix $EA$. *The determinants of $A$ and $EA$ are directly related to the areas of these parallelograms.
Change the matrix $E$ below to several different examples of elementary matrices to give a geometric reason as to why the following three facts are true. *If $E$ is formed by interchanging two rows, then we have $|EA|=-|A|$ (the sign changes), *If $E$ is formed by multiplying a row by a nonzero constant $k$, then $|EA|=k|A|$. *If $E$ is formed by adding a multiple of a row to another, then $|EA|=|A|$ (no change). To answer this one, look for parallel lines.
<sagecell> E=matrix([ [1,0], [1/2,1] ])
A=matrix([ [1,2], [-3,4] ])
c1,c2=A.rows() d1,d2=(E*A).rows() z=(0,0)
boxplot=arrow(z,c1,color='red',aspect_ratio=1) boxplot+=arrow(z,c2,color='blue') boxplot+=arrow(c2,c2+c1,color='red',linestyle='dashed') boxplot+=arrow(c1,c1+c2,color='blue',linestyle='dashed')
boxplot2=arrow(z,d1,color='red',aspect_ratio=1) boxplot2+=arrow(z,d2,color='blue') boxplot2+=arrow(d2,d2+d1,color='red',linestyle='dashed') boxplot2+=arrow(d1,d1+d2,color='blue',linestyle='dashed')
table([ ["Matrix A",A], ["Elementary Matrix E",E], ["Product EA",E*A], ])
table([[ boxplot,boxplot+boxplot2 ]]) </sagecell>