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The question involves several operations with matrices, including addition, scalar multiplication, matrix multiplication, finding the inverse, and calculating the determinant. Let's break down the steps based on the given matrices.

The matrices provided are:

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

### Questions:

1. **Matrix Addition**: Find \( C + D^T \)
   - First, transpose matrix \( D \), then add it to matrix \( C \).

2. **Scalar Multiplication**: Find \( \frac{1}{3} C \)
   - Multiply each element of matrix \( C \) by \( \frac{1}{3} \).

3. **Matrix Multiplication**: Find \( AB \)
   - Multiply matrix \( A \) by matrix \( B \). Since \( A \) is \( 1 \times 2 \) and \( B \) is \( 2 \times 2 \), this operation is valid.

4. **Inverse of Matrix**: Find \( B^{-1} \)
   - Calculate the inverse of the \( 2 \times 2 \) matrix \( B \).

5. **Determinant of Matrix**: Find \( \text{det}(E) \)
   - Compute the determinant of matrix \( E \), which is a \( 3 \times 3 \) matrix.

Let me solve these operations one by one.Here are the results of the matrix operations based on the problem:

1. **Matrix Addition \( C + D^T \)**:
   \[
   \begin{bmatrix} 9 & 5 & 0 \\ 13 & 4 & -14 \end{bmatrix}
   \]

2. **Scalar Multiplication \( \frac{1}{3} C \)**:
   \[
   \begin{bmatrix} 2 & 1.33 & 0 \\ 2.67 & 0.67 & -3.33 \end{bmatrix}
   \]

3. **Matrix Multiplication \( AB \)**:
   \[
   \begin{bmatrix} 0 & 7 \end{bmatrix}
   \]

4. **Inverse of Matrix \( B^{-1} \)**:
   \[
   \begin{bmatrix} 0.714 & 0.429 \\ 0.143 & 0.286 \end{bmatrix}
   \]

5. **Determinant of Matrix \( E \)**:
   \[
   \text{det}(E) = -2
   \]

Would you like further details on any of these results?

Here are 5 related questions for further exploration:
1. What are the properties of matrix transposition and how does it affect matrix addition?
2. How is scalar multiplication applied to a matrix, and what impact does it have on its determinant?
3. How does matrix multiplication work for non-square matrices like \( A \times B \)?
4. What is the geometric interpretation of a matrix inverse?
5. How can we compute determinants for larger matrices using row reduction or other methods?

**Tip:** Always check matrix dimensions before performing operations like multiplication or addition to ensure they are valid.

Perhatikan matriks berikut: A = [1  2], B = [2 -3; -1 5], C = [6 4 0; 8 2 -10], D = [3 5; 1 2; 0 -4], E = [1 2 1; 2 3 4; 1 2 3]. Maka tentukan: 1. Hasil dari matriks C + D^T adalah ..., 2. Hasil dari perkalian skalar 1/3 C adalah ..., 3. Hasil perkalian matriks AB adalah ..., 4. Nilai invers matriks B^-1 adalah ..., 5. Determinan matriks E adalah ....

Learn how to perform various matrix operations including addition, transposition, scalar multiplication, matrix multiplication, and find the inverse and determinant of matrices. This problem covers essential matrix concepts for high school students.

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