What is the necessary condition for a matrix to have eigenvalue 1? What would be the necessary condition for a matrix of any $n \times n$ to have eigenvalue 1? I know that it must have a corresponding eigenvector - that is obvious - I want to know things like how values must be, or things like that. 

The necessary and sufficient condition for $A$ to have eigenvalue $1$ is that $A-I$ is singular.  This is equivalent to any of


*

*$\det(A-I) = 0$

*$\text{Ker}(A-I) \ne \{0\}$

*$\text{Ran}(A-I) \ne {\mathbb F}^n$ (where we're working over the field $\mathbb F$)

*$\text{Ker}(A^T - I) \ne \{0\}$

*$\text{Ran}(A^T - I) \ne  {\mathbb F}^n$

