https://file.aiquickdraw.com/mathbot/file/1726621113289grt0pxgt.jpg

Let's break down the problem from the image you've uploaded:

### Given Matrices:
1. **Matrix A:** \( A = \begin{bmatrix} 1 & 2 \end{bmatrix} \)
2. **Matrix B:** \( B = \begin{bmatrix} 2 & -3 \\ 1 & 5 \end{bmatrix} \)
3. **Matrix C:** \( C = \begin{bmatrix} 6 & 4 & 0 \\ 8 & 2 & -10 \end{bmatrix} \)
4. **Matrix D:** \( D = \begin{bmatrix} 3 & 5 \\ 1 & 2 \\ 0 & -4 \end{bmatrix} \)
5. **Matrix E:** \( E = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 3 \end{bmatrix} \)

The problem is asking for the result of **\( C + D^T \)**, where \( D^T \) is the transpose of matrix \( D \).

### Step 1: Transpose of Matrix \( D \)
Transpose of a matrix is obtained by flipping it over its diagonal. For matrix \( D \), we get:
\[
D^T = \begin{bmatrix} 3 & 1 & 0 \\ 5 & 2 & -4 \end{bmatrix}
\]

### Step 2: Adding \( C \) and \( D^T \)
Now, we can add matrix \( C \) and matrix \( D^T \), element by element:

\[
C = \begin{bmatrix} 6 & 4 & 0 \\ 8 & 2 & -10 \end{bmatrix}, \quad D^T = \begin{bmatrix} 3 & 1 & 0 \\ 5 & 2 & -4 \end{bmatrix}
\]

Performing the element-wise addition:

\[
C + D^T = \begin{bmatrix} 6+3 & 4+1 & 0+0 \\ 8+5 & 2+2 & -10+(-4) \end{bmatrix} = \begin{bmatrix} 9 & 5 & 0 \\ 13 & 4 & -14 \end{bmatrix}
\]

Thus, the result of \( C + D^T \) is:
\[
\boxed{\begin{bmatrix} 9 & 5 & 0 \\ 13 & 4 & -14 \end{bmatrix}}
\]

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Let me know if you need further clarifications or details.

### Here are 5 related questions for further practice:
1. What is the transpose of matrix \( B \)?
2. Can you find the product \( A \times B \)?
3. What is the determinant of matrix \( E \)?
4. How do you find the inverse of matrix \( B \), if it exists?
5. How would you calculate \( D \times E \)?

### Tip:
Always remember that matrix addition requires the matrices to have the same dimensions!

Learn how to compute matrix addition with a transpose operation, using matrices C and D. This problem focuses on matrix transposition and addition techniques, essential for understanding linear algebra. Suitable for students in Grades 10-12.

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