The matrix $A$ in the image is given by:

$A = \begin{bmatrix} -1 & -\sqrt{3} \\ \sqrt{3} & -1 \end{bmatrix}$

Part (i) of the question asks you to calculate the matrix $A^2$, which is the square of the matrix $A$.

Let’s begin by calculating $A^2$. The general formula for squaring a matrix is multiplying the matrix by itself:

$A^2 = A \times A$

Given the matrix:

$A = \begin{bmatrix} -1 & -\sqrt{3} \\ \sqrt{3} & -1 \end{bmatrix}$

Let me now calculate $A^2$.The square of the matrix $A$ is:

$A^2 = \begin{bmatrix} -2 & 3.464 \\ -3.464 & -2 \end{bmatrix}$

Would you like a further breakdown of this calculation or any additional details?

Here are five related questions you might want to explore:

1. What are the eigenvalues and eigenvectors of matrix $A$?
2. Can you calculate the determinant of matrix $A$?
3. What is the trace of matrix $A$?
4. How do you compute the inverse of matrix $A$, if it exists?
5. Is matrix $A$ orthogonal?

Tip: When multiplying matrices, remember that matrix multiplication is not commutative, meaning $A \times B \neq B \times A$ in general.