The Existence of $A^{-1}$ From the Characteristic Polynomial of A

Question

If a matrix $A$ has the characteristic polynomial $p(x)=x^3+x-1$, is it true that $\det(A)×(-1)^n=-1$ where ($n=$the dimension of $A$. Is there any intuition behind this result, assuming that I have it correctly?

Answer 1

I don't know about interpreting the actual value of coefficients of the characteristic polynomial geometrically, (the characteristic polynomial has a very algebraic flavour), but you can definitely see why the constant term affects invertibility.

In your example, if the characteristic polynomial of $A$ is $p(x) = x^3 + x - 1$, then by Cayley-Hamilton we have $A^3 + A - 1 = 0$, which we can rearrange into $A(A^2 + 1) = 1$, and so we find that $A^{-1} = A^2 + 1$. It should be clear we can do this whenever the constant term is nonzero.