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To determine whether the matrix-vector product \( A \cdot x \) is defined or undefined, we must check whether the number of columns in matrix \( A \) matches the number of rows in vector \( x \).

Here’s the procedure for each problem:

1. **Problem 1:**
   - Matrix \( A \) is \( 3 \times 3 \), meaning it has 3 rows and 3 columns.
   - Vector \( x \) is a \( 3 \times 1 \) column vector.
   - **Conclusion**: Since the number of columns in \( A \) (3) matches the number of rows in \( x \) (3), the product \( A \cdot x \) is **defined**.

2. **Problem 2:**
   - Matrix \( A \) is \( 2 \times 3 \), meaning it has 2 rows and 3 columns.
   - Vector \( x \) is a \( 4 \times 1 \) column vector.
   - **Conclusion**: The number of columns in \( A \) (3) does not match the number of rows in \( x \) (4), so the product \( A \cdot x \) is **undefined**.

3. **Problem 3:**
   - Matrix \( A \) is \( 4 \times 4 \), meaning it has 4 rows and 4 columns.
   - Vector \( x \) is a \( 3 \times 1 \) column vector.
   - **Conclusion**: The number of columns in \( A \) (4) does not match the number of rows in \( x \) (3), so the product \( A \cdot x \) is **undefined**.

4. **Problem 4:**
   - Matrix \( A \) is \( 3 \times 5 \), meaning it has 3 rows and 5 columns.
   - Vector \( x \) is a \( 4 \times 1 \) column vector.
   - **Conclusion**: The number of columns in \( A \) (5) does not match the number of rows in \( x \) (4), so the product \( A \cdot x \) is **undefined**.

5. **Problem 5:**
   - Matrix \( A \) is \( 1 \times 5 \), meaning it has 1 row and 5 columns.
   - Vector \( x \) is a \( 6 \times 1 \) column vector.
   - **Conclusion**: The number of columns in \( A \) (5) does not match the number of rows in \( x \) (6), so the product \( A \cdot x \) is **undefined**.

Let me know if you need further explanations or want more details.

Here are 5 related questions:
1. How can the dimensions of a matrix affect matrix multiplication?
2. What conditions must be satisfied for two matrices to be multiplied?
3. Can a matrix multiplication result in a vector? Under what conditions?
4. What are some practical applications of matrix-vector multiplication?
5. What happens when you multiply a square matrix with an identity matrix?

**Tip**: Always ensure the inner dimensions match for matrix multiplication to be valid (columns of the first matrix = rows of the second matrix).

Determine whether the product A*x is defined or undefined for different matrices and vectors.

Analyze whether the product of a matrix A and a vector x is defined or undefined based on matrix dimensions. This problem covers key linear algebra concepts such as matrix multiplication and dimensions. Suitable for advanced high school or college-level students.

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