https://file.aiquickdraw.com/mathbot/file/17265748681106ndter4t.jpg

To find the Cayley-Hamilton theorem for the matrix \( A \) given in the image, we follow the following steps:

### Step 1: Matrix \( A \)
From the image, the matrix \( A \) is:

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

### Step 2: Characteristic Polynomial
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. The characteristic equation of a matrix is given by:

\[
\text{det}(A - \lambda I) = 0
\]

Where \( I \) is the identity matrix and \( \lambda \) represents the eigenvalue.

To calculate the characteristic polynomial, we compute the determinant of \( A - \lambda I \).

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

Now, we'll calculate the determinant of this matrix. I'll proceed with the computation.

The characteristic polynomial of the matrix \( A \) is:

\[
\text{det}(A - \lambda I) = -\lambda^3 + 2\lambda^2 + 2\lambda - 5
\]

### Step 3: Cayley-Hamilton Theorem
According to the Cayley-Hamilton theorem, the matrix \( A \) satisfies its own characteristic equation. Therefore, the characteristic polynomial can be rewritten in terms of the matrix \( A \):

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

This is the matrix equation that \( A \) satisfies, according to the Cayley-Hamilton theorem.

Would you like further details on how to compute powers of the matrix or clarify any part of the process?

Here are 5 related questions to consider:
1. How do we find the eigenvalues and eigenvectors of the matrix \( A \)?
2. What is the inverse of the matrix \( A \), if it exists?
3. How can we use diagonalization to express powers of the matrix \( A \)?
4. How is the Cayley-Hamilton theorem applied in solving systems of differential equations?
5. Can the matrix \( A \) be classified as symmetric, skew-symmetric, or neither?

**Tip:** The Cayley-Hamilton theorem is useful for calculating higher powers of matrices without performing matrix multiplication repeatedly.

Learn how to apply the Cayley-Hamilton theorem to a 3x3 matrix with a characteristic polynomial. Detailed step-by-step explanation including determinant calculation and polynomial derivation. Suitable for undergraduate-level linear algebra courses.

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