The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. To verify the Cayley-Hamilton theorem for a given matrix $A$, we need to follow these steps:

1. Find the characteristic polynomial of $A$.
2. Substitute the matrix $A$ into its own characteristic equation.
3. Verify that the matrix satisfies the characteristic equation.

After verifying the Cayley-Hamilton theorem, we can use it to find the inverse of $A$.

Step 1: Find the characteristic polynomial of $A$

The characteristic polynomial of a matrix $A$ is given by: $p(\lambda) = \det(A - \lambda I)$ where $I$ is the identity matrix of the same size as $A$, and $\lambda$ is the eigenvalue variable.

Let me know the matrix $A$ you are working with, and I will proceed with the verification and find its inverse for you.

Feel free to upload a photo or provide the matrix in text form!