https://file.aiquickdraw.com/mathbot/file/17298855365771ljcm5f1.jpg

The problem in the image asks about matrix operations and their validity given the dimensions of the matrices. Here's how we can evaluate each option:

- **Matrix Dimensions:**
  - \( A \) and \( B \) are both \( 3 \times 8 \) matrices.
  - \( C \) is a \( 7 \times 3 \) matrix.

Now, let’s analyze each option:

### Option A: \( B + C \)
- Matrix addition requires the matrices to have the same dimensions.
- \( B \) is \( 3 \times 8 \), and \( C \) is \( 7 \times 3 \), so they do not have matching dimensions.
- **Not defined.**

### Option B: \( B + A \)
- Both \( B \) and \( A \) are \( 3 \times 8 \) matrices.
- Since their dimensions match, matrix addition is defined.
- **Defined.**

### Option C: \( BC \)
- For matrix multiplication, the number of columns of the first matrix must equal the number of rows of the second matrix.
- \( B \) is \( 3 \times 8 \) and \( C \) is \( 7 \times 3 \), so the number of columns of \( B \) (8) does not match the number of rows of \( C \) (7).
- **Not defined.**

### Option D: \( CB + 2A \)
- First, evaluate \( CB \):
  - \( C \) is \( 7 \times 3 \) and \( B \) is \( 3 \times 8 \).
  - The number of columns of \( C \) (3) matches the number of rows of \( B \) (3), so the matrix product \( CB \) is defined and will have dimensions \( 7 \times 8 \).
- Now, evaluate \( 2A \):
  - \( A \) is \( 3 \times 8 \). The scalar multiplication \( 2A \) will still be a \( 3 \times 8 \) matrix.
  - However, you cannot add a \( 7 \times 8 \) matrix (from \( CB \)) to a \( 3 \times 8 \) matrix (from \( 2A \)), as their dimensions are different.
- **Not defined.**

### Option E: \( CB \)
- As previously discussed in Option D, \( C \) is \( 7 \times 3 \) and \( B \) is \( 3 \times 8 \).
- The matrix multiplication is defined and results in a \( 7 \times 8 \) matrix.
- **Defined.**

### Conclusion:
The valid (defined) options are **B (B + A)** and **E (CB)**.

Let me know if you would like further details or have any questions!

Here are 5 questions to expand on the topic:
1. How can matrix multiplication be generalized for arbitrary dimensions?
2. What are the conditions required for matrix addition?
3. If matrix \( A \) is \( m \times n \) and matrix \( B \) is \( n \times p \), what will be the dimensions of \( AB \)?
4. How does scalar multiplication differ from matrix multiplication?
5. What happens when you multiply a matrix by its transpose?

**Tip:** Always check the dimensions of the matrices involved before performing operations to avoid undefined results.

If A and B are 3 × 8-matrices, and C is a 7 × 3-matrix, which of the following are defined? Check all boxes that apply: A. B + C, B. B + A, C. BC, D. CB + 2A, E. CB

This matrix problem explores the validity of various operations between 3x8 matrices A and B, and a 7x3 matrix C. The operations include matrix addition, scalar multiplication, and matrix multiplication. Suitable for linear algebra students in high school or early college.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation