Suppose that

for some matrix , some vector and some scalar .

Then is called an eigenvector and is the corresponding Eigenvalue.

The thing is, how do you find and ?

Notice if

It might make sense to move everything to one side.

Unfortunately is a matrix and is a scalar. So you instead write

.

Now that the matrices have the same dimensions, you can subtract them

.

Clearly, this has a trivial solution if .

For a nontrivial solution

.

Since this is a zero matrix, it is singular.

So .

We can use this information to find .

In your case:

.

You need to work out

.

Therefore

.

Since this determinant is zero...

You should be able to solve for now.