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What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know something more, please.

Thank you very much.

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If you think about the matrix as representing a linear transformation, then the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed. So for instance, the matrix $2I$ stretches a square of area 1 into a square with area 4, since the determinant is 4. This idea works in all dimensions too, not just 2 or 3!

This also translates well when you get in to more general mappings $f:\Bbb{R}^n\rightarrow\Bbb{R}^n$; if the function is nice enough, you can represent it "locally" by a linear transformation. The (absolute value of the) determinant of this linear transformation gives the "local" volume distortion of the function - i.e. how much the function is stretching or compressing regions of space near a point.

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    $\begingroup$ Also, the sign of the determinant represents whether the transformation is "orientation-preserving." $\endgroup$ – Thomas Andrews Nov 14 '16 at 17:58
  • $\begingroup$ @icurays1 What does it mean when the determinant becomes zero? Does a singular matrix transform the Volume of the space to a single point? $\endgroup$ – v.tralala Feb 2 '20 at 16:33
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    $\begingroup$ @v.tralala if the determinant is zero, one or more eigenvalues is zero, which means that one or more dimensions are being “collapsed”. For instance a 3D volume can collapse to a 2D plane, or to a point. Think about the various types of diagonal matrices with one or more zeros on the diagonal, and how they transform things. $\endgroup$ – icurays1 Feb 2 '20 at 19:18

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