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.