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: