Or, use the definition of "eigenvalue" and "eigenvector".
If you are right that 2 is an eigenvalue, then there must exist a vector,
, x, y, z not all 0, such that
.
That gives the three equations -2x-8 y- 12z= 2x, x+ 4y+ 4z= 2y, and z= 2z.
Of course, that last equations tells us that z= 0 so the first two become -2x- 8y= 2x, or 8y= -4x so that x= -2y, and x+ 4y= 2y and, again, x= -2y. The fact that there exist an infinite number of solutions to those equations confirms that 2
is an eigenvalue. Any eigenvector corresponding to eigenvalue 2 is of the form
.
Now, to find the eigenvectors corresponding eigenvalues 1 and 0, just solve -2x- 8y- 12z= x, x+ 4y= y, z= z, and -2x- 8y- 12z= 0, x+ 4y= 0, z= 0.
On the contrary, eigenvectors are great fun! (Not to mention extremely useful.)