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Interpreting determinants in terms of area

The determinant of a 2X2 matrix tells us what the area of the image of a unit square would be under the matrix transformation. This, in turn, allows us to tell what the area of the image of any figure would be under the transformation. Created by Sal Khan.

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• Does this concept applies as well to higher dimensions? I.e, determinant of 3x3 matrix as a volume of parallelepiped? Does it generalize to higher dimensions? Determinant of 4x4 matrix is ?? of 4D solid?
• Yes, that's correct. The determinant of a 1x1 matrix gives the length of a segment, of a 2x2 the area of a parallelogram, of a 3x3 the volume of a parallelepiped, and of an nxn the hypervolume of an n-dimensional parallelogram.
• What videos should I watch to realize how to get the area of triangles or parallelograms by using vectors on the coordinate plane?
• I don't know of any video specifically addressing it, but hopefully my answer is helpful nonetheless as I had a similar question and decided to solve it by hand. As we know, the area of a parallelogram is basically the base times the height.

It's simple enough to establish the base; use the pythagorean theorem on one of the vectors (preferably the longer).

The height is trickier. You need to know trigonometry and the unit circle for that - particularly sine/cos/tan (and their inverse) in order to calculate the angle between two vectors. Note that the matrix conveniently provides all the details needed to do the arctan operations! So you take the arctan of the vector with the greater angle [1,2] and subtract the arctan of the vector with the smaller angle [3,1] and you're left with the angle between the two vectors.

Once you have the angle, you simply need to set the vector that isn't being used as the base as the hypotenuse (again using the pythagorean theorem to calculate its length). You can then solve for the height by multiplying sine of the angle times the hypotenuse.

Multiply the base and the new-found height and you should then have the area of the parallelogram.

EDIT, this may be useful: https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/proof-matrix-determinant-gives-area-of-image-of-unit-square-under-mapping