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Let's work through the matrix problems from your question systematically. 

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### **Part (a)**
Given:
\[
A = \begin{pmatrix} 4 & 7 \\ 5 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & -2 \\ 4 & -6 \end{pmatrix}
\]
We need to compute the following:

#### (i) \( A - \frac{1}{2} B \)
First, compute \( \frac{1}{2} B \):
\[
\frac{1}{2} B = \frac{1}{2} \begin{pmatrix} 3 & -2 \\ 4 & -6 \end{pmatrix} = \begin{pmatrix} \frac{3}{2} & -1 \\ 2 & -3 \end{pmatrix}
\]
Now subtract:
\[
A - \frac{1}{2} B = \begin{pmatrix} 4 & 7 \\ 5 & 2 \end{pmatrix} - \begin{pmatrix} \frac{3}{2} & -1 \\ 2 & -3 \end{pmatrix} = \begin{pmatrix} 4 - \frac{3}{2} & 7 - (-1) \\ 5 - 2 & 2 - (-3) \end{pmatrix}
\]
\[
A - \frac{1}{2} B = \begin{pmatrix} \frac{5}{2} & 8 \\ 3 & 5 \end{pmatrix}
\]

#### (ii) \( A^T + B \)
First, find the transpose of \( A \), denoted \( A^T \):
\[
A^T = \begin{pmatrix} 4 & 5 \\ 7 & 2 \end{pmatrix}
\]
Now add \( A^T \) to \( B \):
\[
A^T + B = \begin{pmatrix} 4 & 5 \\ 7 & 2 \end{pmatrix} + \begin{pmatrix} 3 & -2 \\ 4 & -6 \end{pmatrix} = \begin{pmatrix} 7 & 3 \\ 11 & -4 \end{pmatrix}
\]

#### (iii) \( A^2 \)
Compute the product \( A \times A \):
\[
A^2 = \begin{pmatrix} 4 & 7 \\ 5 & 2 \end{pmatrix} \times \begin{pmatrix} 4 & 7 \\ 5 & 2 \end{pmatrix}
\]
Using matrix multiplication:
\[
A^2 = \begin{pmatrix} 4 \cdot 4 + 7 \cdot 5 & 4 \cdot 7 + 7 \cdot 2 \\ 5 \cdot 4 + 2 \cdot 5 & 5 \cdot 7 + 2 \cdot 2 \end{pmatrix} = \begin{pmatrix} 48 & 42 \\ 30 & 39 \end{pmatrix}
\]

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### **Part (b)**
Given:
\[
A = \begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix}, \quad B = \begin{pmatrix} 7 & 1 \\ -2 & 9 \\ 4 & 3 \end{pmatrix}
\]
We need to compute the following:

#### (i) \( A \times B \)
Since \( A \) is a \( 2 \times 2 \) matrix and \( B \) is a \( 2 \times 3 \) matrix, the result will be a \( 2 \times 3 \) matrix:
\[
A \times B = \begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix} \times \begin{pmatrix} 7 & 1 \\ -2 & 9 \\ 4 & 3 \endalp

If A = [[4, 7], [5, 2]] and B = [[3, -2], [4, -6]], find:

(i) A - 1/2 B
(ii) A^T + B
(iii) A^2

If A = [[2, 4], [3, 9], [6, 5]] and B = [[7, 1], [-2, 9], [4, 3]], find:

(i) A x B
(ii) B x A

If A = [[x^2, x], [xy, y]], what is the determinant of A, and illustrate whether A has an inverse?

Explore matrix operations, including subtraction, transpose addition, multiplication, and determinants. Solve problems involving A = [[4, 7], [5, 2]] and B = [[3, -2], [4, -6]], and determine the inverse condition for matrices. Suitable for undergraduate students.

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