We are asked to prove the following:

Given a 2x2 matrix:

|a b|

|c d|

Prove or disprove that the two eigenvectors are of the form:

[a-lambda,c]^T and [b,d-lambda]^T

I have been messing around for quite some time now, and I cant see how this works out. I have found the better known one, that if b=/=0 then [b,lamba-a] and if c=/=0 then [lambda-d,c], but this one eludes me. I thought that maybe it couldnt be proved, and that it was false, but all the 2x2 matrices that I have tried it on worked, so I am stumped...