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Let's go through the questions and solve them one by one.

### 1. **State the Cayley-Hamilton theorem.**
The Cayley-Hamilton theorem states that *every square matrix satisfies its own characteristic equation*. In other words, if \( A \) is a square matrix and \( p(\lambda) = \det(\lambda I - A) \) is its characteristic polynomial, then substituting the matrix \( A \) into its characteristic polynomial results in the zero matrix. Formally:
\[
p(A) = 0
\]

### 2. **Verify the Cayley-Hamilton theorem for the matrix A and hence find the inverse of A.**
The matrix given is:
\[
A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}
\]
#### Steps:
- First, find the characteristic polynomial of \( A \).
- Verify that the matrix \( A \) satisfies the polynomial.
- Use the Cayley-Hamilton theorem to find the inverse of \( A \).

I will calculate these steps.

### 3. **Find the inverse of matrix A using minors and cofactors.**
Once we have the inverse of \( A \) from the Cayley-Hamilton method, we can cross-check it with the inverse obtained using the cofactor method.

### 4. **Find the rank of the matrix:**
The matrix given is:
\[
A = \begin{bmatrix} 4 & -2 & 1 \\ 4 & -2 & 2 \\ 0 & -1 & 1 \end{bmatrix}
\]
We can use Gaussian elimination or determinants of minors to find the rank.

### 5. **Give an example of matrices A and B such that \( AB = 0 \), but \( A \neq 0 \) and \( B \neq 0 \).**
This requires us to construct two non-zero matrices \( A \) and \( B \) whose product is zero.

### 6. **Solve using Cramer’s Rule:**
The system of equations is:
\[
\begin{aligned}
5x - 7y + z &= 1 \\
6x - 8y - z &= 15 \\
3x + 2y - 6z &= 7
\end{aligned}
\]
We will solve this system using determinants and Cramer's Rule.

### 7. **Find the Eigenvalues of the matrix \( A = \begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix} \).**
To find the eigenvalues, we need to solve the characteristic equation:
\[
\det(A - \lambda I) = 0
\]

I'll begin solving each problem in sequence and provide details. Let's start with problem 2 by verifying the Cayley-Hamilton theorem for the matrix \( A \) and using it to find the inverse.The characteristic polynomial of matrix \( A \) is:
\[
p(\lambda) = \lambda^3 - 6\lambda^2 + 9\lambda - 4
\]

The eigenvalues of \( A \) are \( \lambda_1 = 1 \), \( \lambda_2 = 4 \), and \( \lambda_3 = 1 \), confirming that the matrix satisfies its characteristic equation according to the Cayley-Hamilton theorem.

To find the inverse of \( A \), I will use the Cayley-Hamilton theorem. The characteristic polynomial gives the relationship:
\[
A^3 - 6A^2 + 9A - 4I = 0
\]
From this, we can express \( A^{-1} \) in terms of \( A \).

Let me now calculate the inverse of \( A \) using the Cayley-Hamilton theorem.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the manual steps for calculating the inverse using the Cayley-Hamilton theorem? I can also help with other parts of the problem.

Here are 5 related questions that might help expand on this:
1. How do you compute the characteristic polynomial of a matrix manually?
2. Can the Cayley-Hamilton theorem be used to find powers of matrices?
3. How does the rank of a matrix relate to its invertibility?
4. What is the geometric interpretation of eigenvalues and eigenvectors?
5. How does Cramer's Rule solve systems of linear equations?

**Tip:** When verifying the Cayley-Hamilton theorem, make sure that the matrix satisfies its own characteristic equation by directly substituting the matrix into the polynomial.

1. State Caley-Hamilton theorem.
2. Verify Caley-Hamilton theorem of the matrix A and hence find inverse of A where A = [[2, -1, 1], [-1, 2, -1], [1, -1, 2]]
3. Find the inverse of matrix A (given in question 2) with the help of minors and cofactors.
4. Find the rank of the matrix A = [[4, -2, 1], [4, -2, 2], [0, -1, 1]]
5. Give an example of matrices A, B such that AB = 0 but A ≠ 0, B ≠ 0.
6. Solve by Cramer's Rule:
5x - 7y + z = 1
6x - 8y - z = 15
3x + 2y - 6z = 7
7. Find the Eigen values of the matrix A = [[5, 4], [1, 2]]

Solve matrix problems involving the Cayley-Hamilton theorem, find the inverse of a matrix using minors and cofactors, determine the rank of a matrix, and solve linear equations with Cramer's Rule. Also, find eigenvalues of a given matrix.

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