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Suppose we are considering a matrix $A\in\Bbb R^{2\times 2}$ withlinearly independent columns. Its determinant $det A$ is the area of the parallelogram enclosed by the two vectors.

Suppose I exchange the rows of the matrix. Then it's determinant becomes $\det A_{transformed}=-\det A$. What is the geometric interpretation of this? I was trying to figure it out by considering that the new parallelogram is flipped along the $x=y$ line, but couldn't go much beyond that.

Can we justify it by saying that flipping the parallelogram along $x=y$ also causes its area vector to flip into the opposite direction?

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  • $\begingroup$ It corresponds to the orientation being reversed. $\endgroup$
    – Nick
    Aug 31, 2016 at 23:51

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The geometrical interpretation for determinant of a $2\times 2$ square matrix is that it is the oriented area of the parallelogram formed by the column vectors.

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