Calculation of PCA

Consider the following data set :

Now we need to calculate the principal component analysis for this data. Here are the eigenvalues and eigenvectors calculated for the covariance matrix of this data :

So the principal component is :

Now , when I have tried to do so by hand . I have found that the eigenvalues are 1.28 and 0.0492 , which are identical to the above solution. surly the principal component corresponds to the eigenvalue = 1.28 . However, when I have tried to solve for the eigenvector, the solution was [[0] [0]] as the augmented matrix of BX=0 Was [[1 0 0] [0 1 0]] . So where is the problem here? and also how can I find the transformed data after I calculate the principle component?

Edit : the covariance matrix is