Show that the equation Ac = 0, where A is a NxN matrix and c is a column matrix with elements c_i, i=1..N can have a non-trivial solution (c != 0) only when det(A) = 0.
inv(A) does not exist when det(A) = 0.
The Attempt at a Solution
If det(A) != 0, we can form inv(A). Then we can write
and so we arrive at the trivial solution. On the contraire, if det(A)=0, we can not form the inverse, and so we can not write
inv(A)*A*c=inv(A)*0, because inv(A) does not exist.
So, there must be other solutions.
Does that make sense? How do I write this in a more formal way?
Thank you for any suggestions