I was reading about determinants and what they actually
represent. I’ve been working with solutions to the Ax = b
problem where A has a determinant which is zero (i.e. it has no
unique inverse). Determinants were always just a thing you did
and there were magic properties about them. I suddenly realise
why intuitively inv(A) exists iff det(A) != 0, simply from the
geometric interpretation of the determinant. The determinant of
an n x n matrix represents the “signed volume”* of an
n-dimensional cube after being multiplied by A (transformed by).
So A is a transformation matrix. If we map to a zero volume, and
thus the determinant is zero, we have zero in one of the
dimensions. Now zero is evil because multiplying everything by
it yields 0. There is no information left regarding the original
dimension as any value would have sufficed to obtain a zero.
Simply speaking, can you inverse 0a = 0? What is a? a is any
value you like, so there is no inverse. Likewise for one of our
dimensions. So, when det(A) = 0 we hav