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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).2010-12-28
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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 1.2010-12-28
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    Thanks for correcting @Rahul, I didn't consider that family of matrices... :D2010-12-28

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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 $\langle(1,0,0),(0,1,0)\rangle$.