What are subspaces of $A$? Can anyone please tell me what are the subspaces of $A$ ?

My Thoughts:  $\{\alpha I + \beta J : \alpha , \beta \in \mathbb R , \beta &gt; 0 \}$ is not a vector space and it does not contain any  non-zero vector spcae as well.
Can anyone help me ?
$A$ is not a set, it's a matrix. Asking for the &quot;subspaces of $A$&quot; probably means the eigenspaces associated to each eigenvalue of $A$. (The question should be clearer about that)
The eigen values of A are $\alpha\pm i\beta$. Since beta&gt;0, the eigen values are distinct and complex conjugates and hence A is invertible. So kernel is trivial and range is whole codomain.