/sci/ - Science & Math

what is the intuition behind [math] \lambda [/math] being an eigenvalue if and only if [math] \det(A-\lambda I)=0 [/math]?

from what i understand:
if [math] f [/math] has a matrix representation [math] A [/math], then [math] \det A [/math] describes the volume of the unit hypercube under the transformation [math] f [/math];
[math] \det \lambda I=\lambda^n [/math] because the linear transformation described by [math] \lambda I [/math] just scales all edges by [math] \lambda [/math];
[math] \det A=0 [/math] means two columns of [math] A [/math] are linearly dependent ("mapped to the same dimension"). so the transformation described by [math] A [/math] sort of "collapses a dimension" of the hypercube; and
if [math] v [/math] is an eigenvector with associated eigenvalue [math] \lambda [/math], then the transformation just scales the all eigenvectors assocaited with [math] \lambda [/math] by [math] \lambda [/math].

what i dont get is:
what [math] \det(A+B) [/math] represents; and
how all of this comes together.

i've tried to add some pictures to help because i dont think ive made all of my points that clear