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How can I prove that a $2\times2$ matrix $A$ is area-preserving iff $\det(A)=1$ or $\det(A)=-1$?

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    You need to check that the area of a parallelogram $ABCD$ is the absolute value of the determinant of the matrix whose columms (or rows) are $\vec{AB}$ and $\vec{AD}$. Then use multiplicativity of det.2011-10-12

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