I felt dumb recently when I noticed that the determinant is sort of “the absolute value for matrices”, considering that it’s literally written using the same signs as the absolute value. Although I guess the determinant of the representation of a complex number as a matrix is ∣∣∣a−bba∣∣∣=a2+b2=|a+bi|2, not |a+bi|. The “signed volume” idea seems related to this, insofar as multiplying a complex number z1 by another z2 will stretch / smush z1 by |z2| (in addition to rotating it).
I felt dumb recently when I noticed that the determinant is sort of “the absolute value for matrices”, considering that it’s literally written using the same signs as the absolute value. Although I guess the determinant of the representation of a complex number as a matrix is ∣∣∣a−bba∣∣∣=a2+b2=|a+bi|2, not |a+bi|. The “signed volume” idea seems related to this, insofar as multiplying a complex number z1 by another z2 will stretch / smush z1 by |z2| (in addition to rotating it).