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The problem requires finding the value of the matrix function \( f(A) \) using the Cayley-Hamilton theorem, given:

\[
f(A) = A^8 - 5A^7 + 7A^6 - 3A^5 + 8A^4 - 5A^3 + 8A^2 - 2A + I
\]

where \( A = \begin{pmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{pmatrix} \).

### Step-by-Step Solution:

1. **Find the characteristic polynomial of matrix \( A \):**

   To apply the Cayley-Hamilton theorem, we first need to compute the characteristic polynomial of matrix \( A \).

   The characteristic polynomial \( p(\lambda) \) of matrix \( A \) is determined by:

   \[
   p(\lambda) = \det(A - \lambda I)
   \]

   where \( I \) is the identity matrix of the same size as \( A \).

2. **Apply the Cayley-Hamilton theorem:**

   The Cayley-Hamilton theorem states that a square matrix satisfies its own characteristic polynomial. So, we can express higher powers of \( A \) (like \( A^8, A^7, \ldots \)) in terms of lower powers of \( A \).

3. **Calculate the required matrix powers:**

   Compute \( A^2, A^3, \ldots \), up to the required power, using the results from the characteristic polynomial and substitution from the Cayley-Hamilton theorem.

4. **Substitute and simplify:**

   Use the characteristic polynomial to substitute the higher-order powers of \( A \) in the expression for \( f(A) \) and simplify to find the result.

Let's proceed with the calculation. I'll start by finding the characteristic polynomial of matrix \( A \).The characteristic polynomial of matrix \( A \) is:

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

### Applying the Cayley-Hamilton Theorem

By the Cayley-Hamilton theorem, matrix \( A \) satisfies its own characteristic polynomial:

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

This equation allows us to express higher powers of \( A \) in terms of \( A^2 \), \( A \), and \( I \).

Now, we will use this characteristic equation to express \( A^8, A^7, \ldots \) in terms of lower powers and substitute these expressions into the function \( f(A) \).

Let's compute the necessary powers of \( A \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the steps manually to compute the function \( f(A) \):

1. **Find matrix powers:** Start by calculating \( A^2 \), \( A^3 \), etc., up to \( A^8 \) using matrix multiplication.

2. **Apply the Cayley-Hamilton theorem:** Use the characteristic polynomial:
   \[
   A^3 = 5A^2 - 7A + 3I
   \]
   to express higher powers of \( A \) (like \( A^4, A^5, \ldots \)) in terms of \( A^2 \), \( A \), and \( I \).

3. **Substitute into \( f(A) \):** Finally, plug these expressions back into the formula for \( f(A) \) and compute it.

Would you like more detailed steps on any part of this process?

### Here are 5 related questions to explore further:
1. How do we find the inverse of a 3x3 matrix?
2. What are the eigenvalues and eigenvectors of matrix \( A \)?
3. How does the Cayley-Hamilton theorem help in matrix calculations?
4. What are the applications of characteristic polynomials in linear algebra?
5. How can we compute matrix exponentials using Taylor series?

**Tip:** Remember that the Cayley-Hamilton theorem is a powerful tool to simplify computations involving high powers of matrices!

Using Cayley-Hamilton Theorem, find the value of the matrix function f(A) = A^8 - 5A^7 + 7A^6 - 3A^5 + 8A^4 - 5A^3 + 8A^2 - 2A + I, where A = [[2, 1, 1], [0, 1, 0], [1, 1, 2]]

This problem involves finding the value of a matrix function using the Cayley-Hamilton Theorem. The function f(A) = A^8 - 5A^7 + 7A^6 - 3A^5 + 8A^4 - 5A^3 + 8A^2 - 2A + I is computed for the matrix A = [[2, 1, 1], [0, 1, 0], [1, 1, 2]].

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