Suppose $A$ is a linear transformation from $R^3$ to $R^3$ and $|det(A)| = 1$. I know that $A$ is volume preserving, but is it also area preserving? For example, if $a$ and $b$ are two vectors in $R^3$ that span a parallelogram, is the area of this parallelogram equal to the area of the paralellogram spanned by $A(a)$ and $A(b)$?
Thank you!