The book I am studying has the definition of a characteristic function as follows.
Let $A\subseteq{X}$. Then
$$\chi_A(x) = \begin{cases} 1, & \text{if $x\in{A}$} \\ 0, & \text{if $x\in{X-A}$} \\ \end{cases} $$
The two cases where I know a characteristic function has been defined is in Linear Algebra where the characteristic function of a matrix is $\det(\lambda{I}-A)=0$ and in Differential Equations where the characteristic function of a linear nth order ordinary differential equation with constant coefficients is determined by letting $r^i=\frac{d^iy}{dt^i}$ and setting the polynomial equal to zero. My questions are these;
1) In the linear algebra case, what would the space $X$ be? by looking at the function defined above I'm assuming it's the set of all square matrices but then what is $A$? And under what conditions do we need to know when $\det(\lambda{I}-A)=1$? Or would it be instead $\det(A)=1$ implying something to do with the $SL$ group? I'm confused.
2) how does this set theoretic definition above help explain roots of nth order linear ode's wiht constant coefficients?