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Answer everything following the instructions

In the image you uploaded, there appear to be multiple questions related to matrix algebra, specifically matrix multiplication and transformation. The instructions suggest reconstructing matrices by following certain rules. Let's walk through each part.

### 1. General Matrix Multiplication Rules
- The questions ask for matrix multiplication and transformations. When multiplying matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
  
### 2. Reconstructing Matrices
The problem also involves reconstructing matrices by following the examples.

Let's go step-by-step to work through the solution:

#### 1. Problem a)
You are given two matrices \( A \) and \( B \):
- \( A = \begin{pmatrix} 2 & 3 \\ 0 & -1 \end{pmatrix} \)
- \( B = \begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix} \)

To multiply these matrices \( A \times B \):
\[
A \times B = \begin{pmatrix} 2 & 3 \\ 0 & -1 \end{pmatrix} \times \begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}
\]

We perform the matrix multiplication by taking the dot product of rows from \( A \) and columns from \( B \):
\[
= \begin{pmatrix} (2 \times -1 + 3 \times 3) & (2 \times 2 + 3 \times 4) \\ (0 \times -1 + -1 \times 3) & (0 \times 2 + -1 \times 4) \end{pmatrix}
\]
\[
= \begin{pmatrix} (-2 + 9) & (4 + 12) \\ (0 - 3) & (0 - 4) \end{pmatrix}
= \begin{pmatrix} 7 & 16 \\ -3 & -4 \end{pmatrix}
\]

#### 2. Problem b)
Similar to part (a), we have:
- \( C = \begin{pmatrix} 1 & -2 \\ 4 & 0 \end{pmatrix} \)
- \( D = \begin{pmatrix} 2 & 3 \\ 5 & 6 \end{pmatrix} \)

To calculate \( C \times D \):
\[
C \times D = \begin{pmatrix} 1 & -2 \\ 4 & 0 \end{pmatrix} \times \begin{pmatrix} 2 & 3 \\ 5 & 6 \end{pmatrix}
\]
We apply the matrix multiplication:
\[
= \begin{pmatrix} (1 \times 2 + -2 \times 5) & (1 \times 3 + -2 \times 6) \\ (4 \times 2 + 0 \times 5) & (4 \times 3 + 0 \times 6) \end{pmatrix}
\]
\[
= \begin{pmatrix} (2 - 10) & (3 - 12) \\ (8 + 0) & (12 + 0) \end{pmatrix}
= \begin{pmatrix} -8 & -9 \\ 8 & 12 \end{pmatrix}
\]

#### 3. Problem c)
Similarly:
- \( E = \begin{pmatrix} 0 & 1 \\ -1 & 3 \end{pmatrix} \)
- \( F = \begin{pmatrix} 2 & 1 \\ 0 & 4 \end{pmatrix} \)

We compute \( E \times F \):
\[
E \times F = \begin{pmatrix} 0 & 1 \\ -1 & 3 \end{pmatrix} \times \begin{pmatrix} 2 & 1 \\ 0 & 4 \end

Matrix multiplication and transformations involving matrices A, B, C, D, E, and F, with operations focusing on reconstructing matrices and applying matrix multiplication rules.

Explore detailed solutions for multiplying matrices A, B, C, D, E, and F, involving matrix transformation rules and multiplication techniques. This linear algebra problem covers essential concepts such as matrix multiplication and matrix structure reconstruction. Suitable for undergraduate students studying Linear Algebra.

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