https://file.aiquickdraw.com/mathbot/file/1726667206400qc7r4d77.jpg

The image you provided shows a matrix \( A = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 2 & 0 \\ 2 & -1 & 0 \end{pmatrix} \).

In reference to the Cayley-Hamilton theorem, it states that every square matrix satisfies its own characteristic equation. To determine if this matrix satisfies the Cayley-Hamilton theorem, we would need to:

1. Find the characteristic polynomial of matrix \( A \).
2. Substitute the matrix \( A \) into this polynomial to check if it satisfies the equation.

Would you like me to proceed with these steps to verify the theorem? Let me know if you want further details.

Here are 5 relative questions:
1. What is the characteristic polynomial of matrix \( A \)?
2. How do you compute the determinant of a 3x3 matrix?
3. What is the eigenvalue of a matrix, and how does it relate to the Cayley-Hamilton theorem?
4. Can you find the inverse of matrix \( A \) if it exists?
5. How can you use the adjugate of a matrix in matrix calculations?

**Tip:** The characteristic equation is derived from the determinant of \( A - \lambda I \), where \( \lambda \) represents an eigenvalue, and \( I \) is the identity matrix.

Explore how to verify the Cayley-Hamilton theorem for the given 3x3 matrix A. This involves calculating the characteristic polynomial and checking if the matrix satisfies its own characteristic equation. Suitable for undergraduate students.

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