Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?
The determinant measures how the volume of the unit box changes. Unit box here means all the points {(a,b,c...,d) | 0<= a,b, ..d <=1 Determinant zero means that it gets squished into smaller dimenisions: eg, for 2x2, the unit square gets sent to a line segment, in 3x3 the unit cube gets sent to either a 2-d or 1-d figure you can't undo these operations, because infinitely many points get sent to the same place. eg |1 0| |0 0| sends all the points with the same y coordinate to the same place, and it squashes the unit square to the unit interval. Is that ok? That's the geometry, we can talk algebraic reasons too.
A very good "intuitive reason" is that det(AB)= det(A)det(B). If AB= I then det(A)det(B)= 1 not 0 so neither det(A) nor det(B) can be 0.
To find a inverse matrix, you must take 1/det. If your det is equal to zero, it is undifined. Paden Roder