in a 3x3 matrix, if the sum of the columns is zero, why is it not invertible?

Question

solution should use the definition of the inverse

Answer 1

If the sum of the columns of matricx A is zero then (c3) = -(c1) + (c2).
ie it is a formulation of dependent vectors
So it is not a one to one transformation and so it does not have a unique inverse!!

ie A^-1 does not exist.

Answer 2

If A is the matrix and it was invertible, let B be its inverse then AB=BA=I (the identity matrix), so for any x: BAx= x. you can choose x = (1,1,1) so that Ax=0, and BAx=0, but x is not 0.

Answer 3

Try the matrix on the vector (1,1,1) You'll get zero.