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Q: Why are determinants defined the weird way they are?
The determinant has a lot of tremendously useful properties, but it’s a weird operation. Here come some properties:1) , if any pair of the vectors are the same, because that corresponds to the parallelepiped being flat. 2) , which is just a fancy math way of saying that doubling the length of any of the sides doubles the volume. By using these properties we can see that switching two vectors in the determinant swaps the sign. For example: The determinant of the matrix is the same as the area of this parallelogram, by definition.
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