I see the exercise you're working on. Let's tackle it step by step. I'll start with Exercise 3.2 (Part 1):

1. Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, and $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$.

Find each of the following:

(i) $A + B$:

$A + B = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} + \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix} 2+1 & 4+3 \\ 3-2 & 2+5 \end{bmatrix} = \begin{bmatrix} 3 & 7 \\ 1 & 7 \end{bmatrix}$

(ii) $A - B$:

$A - B = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix} 2-1 & 4-3 \\ 3-(-2) & 2-5 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 5 & -3 \end{bmatrix}$

(iii) $3A - C$:

$3A = 3 \times \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} 6 & 12 \\ 9 & 6 \end{bmatrix}$ $3A - C = \begin{bmatrix} 6 & 12 \\ 9 & 6 \end{bmatrix} - \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 6 - (-2) & 12 - 5 \\ 9 - 3 & 6 - 4 \end{bmatrix} = \begin{bmatrix} 8 & 7 \\ 6 & 2 \end{bmatrix}$

(iv) $AB$ (matrix multiplication):

$AB = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} \times \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$ Calculating the product: $AB = \begin{bmatrix} 2 \times 1 + 4 \times (-2) & 2 \times 3 + 4 \times 5 \\ 3 \times 1 + 2 \times (-2) & 3 \times 3 + 2 \times 5 \end{bmatrix} = \begin{bmatrix} 2 - 8 & 6 + 20 \\ 3 - 4 & 9 + 10 \end{bmatrix} = \begin{bmatrix} -6 & 26 \\ -1 & 19 \end{bmatrix}$

(v) $BA$:

$BA = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix} \times \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$ Calculating the product: $BA = \begin{bmatrix} 1 \times 2 + 3 \times 3 & 1 \times 4 + 3 \times 2 \\ -2 \times 2 + 5 \times 3 & -2 \times 4 + 5 \times 2 \end{bmatrix} = \begin{bmatrix} 2 + 9 & 4 + 6 \\ -4 + 15 & -8 + 10 \end{bmatrix} = \begin{bmatrix} 11 & 10 \\ 11 & 2 \end{bmatrix}$

This completes Part 1.

Would you like to go over any of these details or have specific questions?

Here are 5 related questions to deepen understanding:

1. Can you explain the process of matrix addition in your own words?
2. How does scalar multiplication, such as $3A$, work in a matrix?
3. Why is matrix multiplication non-commutative (i.e., $AB \neq BA$)?
4. What is the significance of the diagonal elements in matrix multiplication?
5. How would you verify the result of $A - B$?

Tip: Always ensure that the dimensions of matrices are compatible when performing matrix multiplication, which requires the number of columns in the first matrix to match the number of rows in the second matrix.