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The image you've provided shows a matrix \( A \) multiplied by a scalar 2. To verify the Cayley-Hamilton theorem, we need to follow these steps:

1. **Identify the Matrix \( A \):**

   The matrix \( A \) in the image is:
   \[
   A = \begin{pmatrix} 
   3 & 2 & 1 \\ 
   0 & 2 & 0 \\ 
   1 & 2 & 3 
   \end{pmatrix}
   \]

2. **Cayley-Hamilton Theorem:**

   The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. This means that if \( p(\lambda) \) is the characteristic polynomial of matrix \( A \), then substituting the matrix \( A \) into the polynomial will result in the zero matrix.

3. **Characteristic Polynomial:**

   To find the characteristic polynomial \( p(\lambda) \) of \( A \), we calculate the determinant of \( \lambda I - A \), where \( I \) is the identity matrix of the same size as \( A \).

   \[
   p(\lambda) = \det(\lambda I - A)
   \]
   
   This requires calculating the determinant of the matrix \( \lambda I - A \), which is:
   \[
   \lambda I - A = \begin{pmatrix} 
   \lambda - 3 & -2 & -1 \\ 
   0 & \lambda - 2 & 0 \\ 
   -1 & -2 & \lambda - 3 
   \end{pmatrix}
   \]

   The characteristic polynomial is the determinant of this matrix.

4. **Substitute into the Matrix:**

   Once we have the characteristic polynomial, we substitute \( A \) into this polynomial to verify if the resulting matrix is the zero matrix.

Would you like to proceed with the detailed calculations to verify the Cayley-Hamilton theorem for this matrix?

### Relative Questions:
1. How is the characteristic polynomial of a matrix calculated?
2. What is the determinant of a matrix?
3. Can the Cayley-Hamilton theorem be applied to any square matrix?
4. What is the significance of the identity matrix in finding the characteristic polynomial?
5. How do you find the inverse of a matrix?

### Tip:
When working with the Cayley-Hamilton theorem, double-check your characteristic polynomial calculation since any small error can lead to incorrect verification.

Explore how to verify the Cayley-Hamilton theorem for a given matrix A = [3 2 1; 0 2 0; 1 2 3]. Understand the steps involved in calculating the characteristic polynomial and applying the theorem. Suitable for advanced mathematics enthusiasts and students.

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