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Let's go through the exercises step by step. The image contains two main exercises labeled **E1** and **E2**:

### E1. Matrix Calculations

You are required to compute the product of matrices for several sub-questions:

- **a)**  
  \( \begin{pmatrix} 4 & 5 \\ 6 & -1 \end{pmatrix} \times \begin{pmatrix} 2 & 1 \\ -3 & -2 \end{pmatrix} \)

- **b)**  
  \( \begin{pmatrix} 1 & -2 \\ 4 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix} \)

- **c)**  
  \( \begin{pmatrix} -1 & 2 \\ 1 & 0 \\ 2 & i^2 \end{pmatrix} \times \begin{pmatrix} 0 & -2 & 3 \\ 1 & 2 & 3 \\ 1 & -1 & -4 \end{pmatrix} \)

- **d)**  
  \( \begin{pmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix} \times \begin{pmatrix} -1 & -1 & \cos(0) \\ 2 & -1 & \sin\left(\frac{\pi}{2}\right) \\ 0 & 1 & \tan\left(\frac{\pi}{4}\right) \end{pmatrix} \)

- **e)**  
  \( \begin{pmatrix} 1 & 2 & 3 \\ 3 & -1 & 1 \end{pmatrix} \times \begin{pmatrix} -1 & -2 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & -1 \end{pmatrix} \times \begin{pmatrix} 1 & 1 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} \)

### E2. For each matrix pair \( A, B \), calculate:
- \( AB \) (matrix multiplication)
- \( BA \) (matrix multiplication)
- \( A^t B \) (transpose of \( A \), then multiply by \( B \))
- \( B^t A \) (transpose of \( B \), then multiply by \( A \))

#### Sub-questions:
- **a)**  
  \( A = \begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix}, B = \begin{pmatrix} \frac{1}{2} & 1 \\ 1 & -\frac{1}{2} \end{pmatrix} \)

- **b)**  
  \( A = \begin{pmatrix} 2 \\ 3 \end{pmatrix}, B = \begin{pmatrix} -3 & 1 \\ -1 \end{pmatrix} \)

- **c)**  
  \( A = \begin{pmatrix} -1 & 1 & \sin\left(\frac{\pi}{6}\right) \\ \cos\left(\frac{\pi}{2}\right) & 1 & 2 \end{pmatrix}, B = \begin{pmatrix} -2 & 1 \\ 1 & 3 \\ -2 & \tan(0) \end{pmatrix} \)

- **d)**  
  \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}, B = \begin{pmatrix} 0 & 0 & -4 \\ -3 & 0 & 0 \\ 1 & -1 & 1 \end{pmatrix} \)

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Matrix operations and transformations, including matrix multiplication, trigonometric matrix elements, and calculating transposes of matrices.

Solve complex matrix operations, including matrix multiplication, transposes, and the application of trigonometric functions to matrix elements. This problem involves linear algebra concepts and trigonometric identities, making it suitable for advanced high school or undergraduate math students.

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