https://file.aiquickdraw.com/mathbot/file/1726621576014p3wyonus.jpg

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.

Based on the matrices A, B, C, D, and E provided in the image, solve the following: 1. Result of C + D^T, 2. Result of scalar multiplication 1/3 * C, 3. Result of matrix multiplication AB.

Explore detailed solutions for matrix operations including addition of C and D^T, scalar multiplication of 1/3 with C, and the multiplication of matrices A and B. Suitable for university level studies in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation