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The problem requires various matrix operations using the provided matrices \( A \), \( B \), \( C \), \( D \), and \( E \). Let's break down the tasks and perform the calculations:

### Given matrices:
- \( A = \begin{pmatrix} -2 & 5 & -5 \\ 3 & -4 & 3 \end{pmatrix} \)
- \( B = \begin{pmatrix} 3 & -1 & 9 \\ 11 & 6 & 4 \end{pmatrix} \)
- \( C = \begin{pmatrix} -3 & 10 \\ 4 & -2 \end{pmatrix} \)
- \( D = \begin{pmatrix} 11 & -1 \\ -2 & 3 \\ 4 & 10 \end{pmatrix} \)
- \( E = \begin{pmatrix} 9 & 2 & 11 \\ 2 & 13 & -4 \\ -5 & -6 & 2 \end{pmatrix} \)

### Operations:
#### a. \( 3A + B \)
We'll calculate \( 3A \) and add \( B \).

#### b. \( AD + 2C \)
This involves matrix multiplication \( AD \) and adding twice of matrix \( C \).

#### c. \( B^T C \)
Transpose matrix \( B \), then multiply it with matrix \( C \).

#### d. \( (A + B)^T \)
Add matrices \( A \) and \( B \), then find the transpose of the result.

#### e. \( AE + B \)
Multiply matrices \( A \) and \( E \), then add matrix \( B \).

I will start calculating these operations one by one.Here are the results of the matrix operations:

#### a. \( 3A + B \)
\[
3A + B = \begin{pmatrix} -3 & 14 & -6 \\ 20 & -6 & 13 \end{pmatrix}
\]

#### b. \( AD + 2C \)
\[
AD + 2C = \begin{pmatrix} -58 & -13 \\ 61 & 11 \end{pmatrix}
\]

#### c. \( B^T C \)
\[
B^T C = \begin{pmatrix} 35 & 8 \\ 27 & -22 \\ -11 & 82 \end{pmatrix}
\]

#### d. \( (A + B)^T \)
\[
(A + B)^T = \begin{pmatrix} 1 & 14 \\ 4 & 2 \\ 4 & 7 \end{pmatrix}
\]

#### e. \( AE + B \)
\[
AE + B = \begin{pmatrix} 20 & 90 & -43 \\ 15 & -58 & 59 \end{pmatrix}
\]

Would you like further details on any of these calculations or need clarification?

Here are some related questions to deepen your understanding:
1. How do you calculate the transpose of a matrix, and why is it important in matrix multiplication?
2. What properties must matrices satisfy to perform matrix multiplication?
3. How does scalar multiplication of a matrix affect its elements?
4. Can the order of multiplication of matrices \( A \) and \( B \) be swapped? Why or why not?
5. What is the difference between matrix addition and matrix multiplication?

**Tip:** Remember that matrix multiplication is not commutative, meaning \( A \times B \neq B \times A \) in general.

Determine the following operations using the provided matrices A, B, C, D, and E:
 a. 3A + B
 b. AD + 2C
 c. B^T C
 d. (A + B)^T
 e. AE + B

Learn how to perform matrix operations such as scalar multiplication, matrix addition, and matrix transpose. This problem covers calculations involving 3A + B, AD + 2C, B^T C, and more. Suitable for Grades 10-12.

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