Cayley Hamilton Theorem Inverse Calculation

Question

How can I find the inverse of $A$ using Cayley Hamilton Theorem?

    A=    0 1 0 0
          0 0 1 0
          0 0 0 1
          1 0 0 0

The Characteristic equation of $A$, I get is $A^4=0$, which implies $A=0$ which is clearly not true. Please help.

Answer 1

I'll expand my above comment. Suppose we have a $4$-dimensional square matrix $A$ (which is not the zero matrix) with characteristic polynomial $p(\lambda)=\lambda^4$; as I previously said if $B$ is a matrix, $B^n=0$ does not imply in general that $B$ is the zero matrix. Indeed, Cayley-Hamilton itself provides examples of this fact: take our matrix $A$, we have $p(A)=A^4=0$, while $A$ is non-zero (perform a direct calculation if you're sceptic).

Answer 2

Expanding $|xI-A|$ along the first row gives $$x(x^3)+1(-1)=x^4-1$$ for the characteristic polynomial. Thus, $$A^4-I=O,$$ $$\text{so that }\ \ \ \ \ A^{-1}=A^3.$$