Matrix X has r rows and c columns, and matrix Y has c rows and d columns, where r, c and d are different. Which of the follwing statements must be true?
I. The product YX exists.
II. The product of XY exists and has r rows and d columns.
III. the product XY exists and has c rows and c columns.

Options

A. I only
B. II only
C. III only
D. I and II

Explanation

To solve the problem, we need to analyze the statements regarding the multiplication of matrices X and Y based on their dimensions.

1. **Understanding the dimensions of the matrices:**
- Matrix X has dimensions \( R \times C \) (R rows and C columns).
- Matrix Y has dimensions \( C \times D \) (C rows and D columns).

2. **Analyzing Statement I: The product YX exists.**
- For the product \( YX \) to exist, the number of columns in Y must equal the number of rows in X.
- Matrix Y has \( D \) columns and matrix X has \( R \) rows.
- Since \( D \) (columns of Y) is not equal to \( R \) (rows of X), the product \( YX \) does not exist.
- Therefore, **Statement I is false**.

3. **Analyzing Statement II: The product XY exists and has R rows and D columns.**
- For the product \( XY \) to exist, the number of columns in X must equal the number of rows in Y.
- Matrix X has \( C \) columns and matrix Y has \( C \) rows.
- Since they are equal, the product \( XY \) exists.
- The resulting matrix will have dimensions equal to the number of rows in X and the number of columns in Y, which is \( R \times D \).
- Therefore, **Statement II is true**.

4. **Analyzing Statement III: The product XY exists and has C rows and C columns.**
- As established, the product \( XY \) exists and has dimensions \( R \times D \).
- Since \( R \) and \( D \) are different from \( C \), the resulting matrix does not have \( C \) rows and \( C \) columns.
- Therefore, **Statement III is false**.

5. **Conclusion:**
- The only statement that is true is Statement II. Thus, the correct answer is that only Statement II is correct.

### Final Answer:
- **Only Statement II is correct.**