Let's solve the questions step by step based on the provided matrices in the image.

Matrices given:

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

1. Find $C + D^T$

Here, we need to calculate the transpose of matrix $D$ and then add it to matrix $C$.

• Transpose of $D$, $D^T$, is: $D^T = \begin{bmatrix} 3 & 1 & 0 \\ 5 & 2 & -4 \end{bmatrix}$

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

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

2. Find $\frac{1}{3} C$

Multiply each element of matrix $C$ by $\frac{1}{3}$:

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

3. Find $A \times B$

Multiply matrix $A$ by matrix $B$: $A \times B = \begin{bmatrix} 1 & 2 \end{bmatrix} \times \begin{bmatrix} 2 & -3 \\ 1 & 5 \end{bmatrix}$

Matrix multiplication: $A \times B = \begin{bmatrix} (1 \times 2 + 2 \times 1) & (1 \times -3 + 2 \times 5) \end{bmatrix}$

This results in: $A \times B = \begin{bmatrix} 4 & 7 \end{bmatrix}$

Would you like any detailed explanation on any of these steps, or do you have further questions?

Questions for further exploration:

1. What happens if we calculate the determinant of matrix $B$?
2. Can matrix $C$ be multiplied by matrix $E$? Why or why not?
3. What is the inverse of matrix $B$, if it exists?
4. How would we compute $D \times E$?
5. Can matrix $D^T \times C$ be calculated? If so, what is the result?

Tip:

Remember, matrix multiplication is not commutative, meaning $A \times B \neq B \times A$ in most cases.