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The question asks: 

**If A is an \( m \times n \) matrix and B is an \( l \times p \) matrix, then the product BA will exist if:**

- (A) \( m = n \)
- (B) \( l = p \)
- (C) \( n = p \)
- (D) \( m = p \)

### Solution:

For the matrix multiplication \( BA \) to be defined, the number of columns in \( B \) must match the number of rows in \( A \).

- Matrix \( B \) is of size \( l \times p \), which means it has **p columns**.
- Matrix \( A \) is of size \( m \times n \), which means it has **n rows**.

Thus, the condition for the product \( BA \) to exist is that the number of columns of \( B \), \( p \), must equal the number of rows of \( A \), \( n \).

Therefore, the correct answer is **(C) \( n = p \)**.

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Would you like further explanation or have any questions? Here are some related questions for practice:

1. If \( A \) is \( 2 \times 3 \) and \( B \) is \( 3 \times 4 \), can you multiply \( AB \)?
2. If \( A \) is \( 4 \times 5 \) and \( B \) is \( 5 \times 6 \), can you multiply \( BA \)?
3. What condition must be satisfied for the product \( AB \) of matrices to be defined?
4. If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times p \) matrix, what will be the size of the product \( AB \)?
5. What is the identity matrix and how does it affect matrix multiplication?

**Tip**: Always check the dimensions of matrices before attempting multiplication.

If A is an m x n matrix and B is an l x p matrix, then the product BA will exist if:

Understand the conditions under which the product of two matrices A (m x n) and B (l x p) will exist. This problem covers the fundamental concept of matrix multiplication in linear algebra, explaining why the number of columns in matrix B must match the number of rows in matrix A for BA to be defined. Suitable for Grades 11-12 or early university.

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