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Let's analyze the problem:

- We are given two matrices, \( A \) and \( B \):
  - \( A \) is a \( 3 \times 10 \) matrix.
  - \( B \) is an \( m \times n \) matrix.
  
- We are told that \( B A^T \) results in a \( 21 \times 3 \) matrix.

### Step-by-step reasoning:

1. **Matrix Transpose \( A^T \)**:
   - The matrix \( A \) is \( 3 \times 10 \).
   - The transpose \( A^T \) will therefore be \( 10 \times 3 \).

2. **Multiplication of Matrices**:
   - We are multiplying \( B \) (which is \( m \times n \)) by \( A^T \) (which is \( 10 \times 3 \)).
   - For matrix multiplication to be defined, the number of columns of \( B \) must match the number of rows of \( A^T \). Therefore, **\( n = 10 \)**.
   - The result of multiplying an \( m \times 10 \) matrix by a \( 10 \times 3 \) matrix should result in an \( m \times 3 \) matrix.
   - We are told that the result of \( B A^T \) is a \( 21 \times 3 \) matrix.
   
3. **Conclusion**:
   - For the resulting matrix to be \( 21 \times 3 \), the number of rows of \( B \), i.e., \( m \), must be **21**.

### Correct Answer:
The value of \( m \) is **21**.

Would you like more details or clarification on any part of this solution? Here are some related questions for further practice:

1. How do you determine the dimensions of the product of two matrices?
2. What are the properties of matrix transposition?
3. Can you find an example where multiplying matrices of different dimensions results in a valid product?
4. What is the rank of a matrix and how does it relate to matrix dimensions?
5. How do matrix dimensions affect the determinant of a square matrix?

**Tip**: Always check the dimensions of the matrices involved before attempting matrix multiplication!

Let A be a 3 x 10 matrix, and let B be m x n. Suppose that B * A^T is a 21 x 3 matrix. What is the value of m?

Solve a matrix multiplication problem where B is an m x n matrix and A is a 3 x 10 matrix. Given that B * A^T results in a 21 x 3 matrix, this solution helps find the value of m. Suitable for students in Grades 10-12.

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