To find the inverse of matrix $A$ and compute $A^4$ using the Cayley-Hamilton theorem, we proceed as follows:

1. Cayley-Hamilton Theorem: According to this theorem, every square matrix satisfies its own characteristic polynomial. If $A$ is an $n \times n$ matrix with characteristic polynomial $p(\lambda)$, then $p(A) = 0$.

Step 1: Find the Characteristic Polynomial of $A$

Let $A$ be the given matrix. To find the characteristic polynomial, we compute the determinant of $A - \lambda I$, where $I$ is the identity matrix and $\lambda$ is a scalar: $\text{det}(A - \lambda I) = 0$ This will yield the characteristic polynomial $p(\lambda)$.

Step 2: Apply Cayley-Hamilton Theorem

Substitute $A$ into its characteristic polynomial to get a matrix equation that $A$ satisfies. This equation will be of the form: $A^n + c_{n-1}A^{n-1} + \cdots + c_1A + c_0I = 0$ where $c_0, c_1, \dots, c_{n-1}$ are constants derived from the characteristic polynomial.

Step 3: Solve for $A^{-1}$

Rearrange the matrix equation to isolate terms involving $A$ on one side and solve for $A^{-1}$ in terms of powers of $A$.

Step 4: Compute $A^4$

Using the relation from the Cayley-Hamilton theorem, express $A^4$ in terms of lower powers of $A$.

To proceed, please provide the actual entries of matrix $A$ so I can compute these values specifically.