I must reduce the following matrix to its Jordan normal form: .

My attempt: I first calculate its characteristic polynomial which gave me .

So there are 2 different eigenvalues, namely and .

Now, I think I have to find the solution of .

I did lots of arithmetics to reach that so that it gives me ... Mathematica doesn't agree on , it seems I've made some errors in some values in the last column, but the matrix has the same form and therefore is still worth which doesn't give me anything useful I believe.

I don't know how to proceed.