Or, just apply the definition of "eigenvector".
If 2 is an eigenvalue, then there exist a non-zero eigenvector, such that
That gives the two equations 3x+ y= 2x and x+ 3y= 2y. Both of those reduce to the equation y= -x. That is
.
An eigenvector corresponding to eigenvalue 2 is
With eigenvalue 4 instead of 2, those equations become 3x+ y= 4x and x+ 3y= 4y, both of which reduce to y= x. An eigenvector corresponding to eigenvalue 4 is .