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The problem asks to find \( A^{-1} \) using the **Cayley-Hamilton Theorem** for the matrix \( A \). The matrix \( A \) is given as:

\[
A = \begin{pmatrix}
1 & 0 & 3 \\
2 & 1 & -1 \\
1 & -1 & 3
\end{pmatrix}
\]

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. We will follow these steps:

### 1. **Find the Characteristic Polynomial:**
   The characteristic polynomial is given by:

   \[
   p(\lambda) = \det(A - \lambda I)
   \]
   where \( I \) is the identity matrix, and \( \lambda \) is the eigenvalue.

### 2. **Apply the Cayley-Hamilton Theorem:**
   After finding the characteristic equation, we replace \( \lambda \) by the matrix \( A \) itself. This will result in an expression of the form:

   \[
   A^3 + c_2 A^2 + c_1 A + c_0 I = 0
   \]

   From this, we can solve for \( A^{-1} \).

I'll now compute the inverse of \( A \) using this process.The characteristic polynomial of matrix \( A \) is:

\[
p(\lambda) = \lambda^3 - 5\lambda^2 + 3\lambda + 7
\]

By the **Cayley-Hamilton Theorem**, the matrix \( A \) satisfies its own characteristic equation, so:

\[
A^3 - 5A^2 + 3A + 7I = 0
\]

We can solve for \( A^{-1} \) by rearranging this equation as:

\[
7I = -A^3 + 5A^2 - 3A
\]

Multiplying both sides by \( \frac{1}{7} \), we get:

\[
A^{-1} = \frac{1}{7}(-A^3 + 5A^2 - 3A)
\]

Next, I'll compute \( A^{-1} \) based on this formula.The inverse of the matrix \( A \) is:

\[
A^{-1} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

Thus, \( A \) is the identity matrix itself, and its inverse is also the identity matrix. 

Would you like further details or have any questions about the steps involved?

Here are some related questions for further exploration:

1. How is the Cayley-Hamilton theorem used to compute matrix inverses in general?
2. What is the process of computing the characteristic polynomial manually?
3. Why does the identity matrix serve as its own inverse?
4. How would the procedure differ for a larger or more complex matrix?
5. What are other methods to find the inverse of a matrix besides Cayley-Hamilton?

**Tip**: Always verify the correctness of a matrix inverse by multiplying it with the original matrix to check if the result is the identity matrix.

Using the Cayley-Hamilton theorem, find A^{-1} for the given matrix A.

Explore how to find the inverse of a matrix using the Cayley-Hamilton theorem. This problem involves understanding the characteristic polynomial, applying the Cayley-Hamilton equation, and calculating the matrix inverse. Suitable for college-level students studying linear algebra.

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