How to use a determinant to find transformed area

A linear transformation can stretch, shrink, or flatten a shape. Its determinant tells us the factor by which area changes. Once we know that factor and the original area, we can find the image’s area without calculating every new side length.

Use the absolute value of the determinant

Aimage=|detM|Aoriginal

The absolute value gives a nonnegative area factor. A negative determinant also tells us that the transformation reverses orientation. If the determinant is zero, every planar region has image area zero.

Find the original rectangle’s area

The rectangle has vertices (1,1), (1,6), (-3,6), and (-3,1). Its width is 4 and its height is 5, so the area is 20. Use the transformation matrix

M=[-3140]

Calculate the determinant and apply the scale factor

detM=(-3)(0)-1(4)=-4Aimage=|-4|(20)=80
A rectangle and its transformed parallelogramThe original rectangle has width four and height five, so its area is twenty. The image has horizontal base five and perpendicular height sixteen, so its area is eighty. The panels use different scales.Area 20Area 8045516
The image has four times the original area. The two panels use different scales to fit both shapes.
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