Q:

How to calculate eigenvectors from an identity matrix with eigenvalue $= 1$?

When determining the eigenvectors from matrix $A$:
$$
      \begin{pmatrix}
      1 & 0 \\
      0 & 1 \\
      \end{pmatrix}
$$
I found the eigenvalue to be $ \lambda = 1 $ 
calculating  $ (A - 1*\lambda) $  gives me the matrix :
$$ \left[
    \begin{array}{cc|c}
      0&0&0\\
      0&0&0
    \end{array}
\right] $$
Which eigenvector would this produce?

A:

For $\lambda=1$, the Eigenvector equation is
$$0v=0.$$
This clearly means that any vector is a solution.

By the way, this can be found directly by identifying
$$Av=\lambda v$$ and $$Iv=v.$$

