Varify cayley Hamilton theorem for given matrix

A=  0     1       2
2    -3       0
1     1      -1

The **Cayley-Hamilton theorem** states that every square matrix satisfies its own characteristic equation. In this case, we are given the matrix \( A \):

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

### Steps to verify the Cayley-Hamilton theorem:

1. **Find the characteristic equation of matrix \( A \)**:
   - The characteristic equation is given by:
     \[
     \text{det}(A - \lambda I) = 0
     \]
     where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.

2. **Compute \( A - \lambda I \)**:
   The identity matrix \( I \) is:
   \[
   I = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]
   So, \( A - \lambda I \) is:
   \[
   A - \lambda I = \begin{pmatrix}
   0 - \lambda & 1 & 2 \\
   2 & -3 - \lambda & 0 \\
   1 & 1 & -1 - \lambda
   \end{pmatrix}
   = \begin{pmatrix}
   -\lambda & 1 & 2 \\
   2 & -3 - \lambda & 0 \\
   1 & 1 & -1 - \lambda
   \end{pmatrix}
   \]

3. **Find the determinant of \( A - \lambda I \)**:
   The determinant of a 3x3 matrix is given by:
   \[
   \text{det}(A - \lambda I) = -\lambda \begin{vmatrix}
   -3 - \lambda & 0 \\
   1 & -1 - \lambda
   \end{vmatrix}
   - 1 \begin{vmatrix}
   2 & 0 \\
   1 & -1 - \lambda
   \end{vmatrix}
   + 2 \begin{vmatrix}
   2 & -3 - \lambda \\
   1 & 1
   \end{vmatrix}
   \]
   Let me compute this determinant.The characteristic equation of the matrix \( A \) is:

\[
-\lambda^3 - 4\lambda^2 + \lambda + 12 = 0
\]

This is the characteristic polynomial of \( A \). According to the Cayley-Hamilton theorem, the matrix \( A \) should satisfy this equation, meaning that if we replace \( \lambda \) with \( A \) in the polynomial, we should get the zero matrix.

4. **Verify if the matrix satisfies its own characteristic equation**:
   We need to compute:

\[
A^3 + 4A^2 - A - 12I
\]

I'll compute this expression to check if it results in the zero matrix.It seems like I can't do more advanced data analysis right now. Please try again later.

You can still compute \( A^3 + 4A^2 - A - 12I \) manually, or I can walk you through the steps.

Do you want to proceed with a manual explanation or have any questions?  
Here are 5 related questions to consider:

1. How is the Cayley-Hamilton theorem useful in finding matrix functions?
2. What are eigenvalues and how do they relate to the characteristic equation?
3. How can we compute powers of matrices efficiently?
4. What is the role of the identity matrix in matrix equations?
5. How can the determinant help in solving systems of equations?

**Tip:** Understanding matrix algebra helps in computer graphics and solving linear systems.

Varify Cayley Hamilton theorem for given matrix A = [0, 1, 2; 2, -3, 0; 1, 1, -1]

Verify the Cayley-Hamilton theorem for the matrix A = [0, 1, 2; 2, -3, 0; 1, 1, -1] with detailed steps. This problem explores key concepts in linear algebra, such as the characteristic equation and matrix theory. Suitable for undergraduate students.

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