Let such a matric be A then:

A (0,y)' = (A_{1,2}y, A_{2,2}y)'

so if A (0, y) = (0, y') for some y', then A_{1,2}=0. Also if A_{1,2}=0 then

A (0, y) = (0, y') for some y'.

So the set of 2x2 matrices you seek is the set with the right hand top

corner element equal to zero.

(This is when we consider the matrix acting on a column vector, if we have

row vectors and the action is in the order yA, then the A will be the transpose

of that in the other representation of the linear operator and the set will be

the set with the left hand bottom corner element equal to zero.)

RonL