Proving "AC=CA=0" for 2x2 Matrices

[Mathmos6]

Apologies, the title messed up - was meant to be 'existence of a C: AC=CA=0 for 2x2 matrices'.

Homework Statement

How would one show 'nicely' that for any 2x2 non-zero matrix A, there exists some 2x2 non-zero matrix C such that AC=CA=0? I can see how to show it by showing that the determinant is '0' for both A and C, so when you multiply out component-wise for AC=0, CA=0, you get 8 equations for the components of A/C and I can show that they can all be paired up as 'equivalent' equations (using ad=bc for both matrices) so in fact you have 4 equations in 4 unknowns - the 4 components of c - so they are solvable, so there does exist some appropriate 'c' - however that's clearly a horrible and slow way of proving the result, can anyone suggest a faster or neater method?

Thanks a lot!

 

[Dick]

You are trying to prove something that isn't even true. Suppose A is the identity matrix? Or did you mean to specify that A is singular? In that case it has a zero eigenvector. Use that to construct C.

 

[Mathmos6]

You are trying to prove something that isn't even true. Suppose A is the identity matrix? Or did you mean to specify that A is singular? In that case it has a zero eigenvector. Use that to construct C.

Sorry, there were 2 parts to the question phrased in a very unrelated way but obviously they were related, implying that yes, A is singular. Thanks very much for the help :)