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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!

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Well, if $\det\;\mathbf A=\pm 1$, then the matrix would have to be orthogonal (prove this). –  J. M. Dec 28 '10 at 10:34
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@J.M.: Only if $A$ is orthogonal! A trivial counterexample suffices for the original problem: the diagonal matrix with entries 2, 1/2, and 1. –  Rahul Dec 28 '10 at 10:37
    
Thanks for correcting @Rahul, I didn't consider that family of matrices... :D –  J. M. Dec 28 '10 at 10:51
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up vote 7 down vote accepted

Matrices of the form $\begin{pmatrix}X & 0\\\\0 & \text{det}(X)^{-1}\end{pmatrix}$ with $X$ any invertible 2 by 2 matrix with determinant not equal to $\pm1$ give a host of counter examples: consider the action of such a matrix on a parallelogram in the subspace $\langle(1,0,0),(0,1,0)\rangle$.

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