Let's break down the problem step by step. From the image, we are asked to perform the following:

Given Matrices:

• $A = [1 \quad 2]$
• $B = \begin{bmatrix} 2 & -3 \\ -1 & 5 \end{bmatrix}$
• $C = \begin{bmatrix} 6 & 4 & 0 \\ 8 & 2 & -10 \end{bmatrix}$
• $D = \begin{bmatrix} 3 & 5 \\ 1 & 2 \\ 0 & -4 \end{bmatrix}$
• $E = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 3 \end{bmatrix}$

Questions:

1. Find the result of $C + D^T$, where $D^T$ is the transpose of $D$.
2. Find the result of the scalar multiplication $\frac{1}{3} C$.

I will now solve each question.

1. Finding $C + D^T$

First, we need to compute the transpose of matrix $D$, denoted as $D^T$, and then add it to matrix $C$.

$D^T = \begin{bmatrix} 3 & 1 & 0 \\ 5 & 2 & -4 \end{bmatrix}$

Now, adding this to $C$:

$C + D^T = \begin{bmatrix} 6 & 4 & 0 \\ 8 & 2 & -10 \end{bmatrix} + \begin{bmatrix} 3 & 1 & 0 \\ 5 & 2 & -4 \end{bmatrix}$

Let's calculate that.

2. Scalar Multiplication $\frac{1}{3} C$

We need to multiply each element of $C$ by $\frac{1}{3}$:

$\frac{1}{3} C = \frac{1}{3} \begin{bmatrix} 6 & 4 & 0 \\ 8 & 2 & -10 \end{bmatrix}$

I will now calculate both results.Here are the results for the given tasks:

1. $C + D^T$:

$C + D^T = \begin{bmatrix} 9 & 5 & 0 \\ 13 & 4 & -14 \end{bmatrix}$

2. Scalar multiplication $\frac{1}{3} C$:

$\frac{1}{3} C = \begin{bmatrix} 2 & 1.33 & 0 \\ 2.67 & 0.67 & -3.33 \end{bmatrix}$

Would you like further details or explanations on any step?

Here are 5 related questions to expand on these topics:

1. How is the transpose of a matrix defined, and why do we use it?
2. Can you explain the difference between scalar multiplication and matrix multiplication?
3. What are the properties of matrix addition?
4. How can we check if two matrices are conformable for addition?
5. What are some real-life applications of matrix operations like transposition and scalar multiplication?

Tip: When adding or multiplying matrices, always check that their dimensions are compatible. For addition, both matrices must have the same size.