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The question is asking when the system of linear equations \( A\mathbf{x} = \mathbf{b} \), where \( A \) is a **non-invertible** \( n \times n \) matrix, will have a solution and whether that solution is unique.

### Key Points:
1. **Invertibility**: If \( A \) is invertible, the solution is always unique and can be found as \( \mathbf{x} = A^{-1}\mathbf{b} \). However, if \( A \) is non-invertible (singular), we cannot directly compute \( A^{-1} \), and other considerations arise.
2. **Non-invertible Case**: For a non-invertible matrix \( A \), the system \( A\mathbf{x} = \mathbf{b} \) may or may not have solutions, depending on \( \mathbf{b} \).
   - **Solution Existence**: Solutions exist if \( \mathbf{b} \) lies in the column space of \( A \), meaning that \( \mathbf{b} \) can be expressed as a linear combination of the columns of \( A \). 
   - **Unique Solution**: If \( A \) is non-invertible, the system might either have **no solution**, **one solution**, or **infinitely many solutions**. In the case where there is a solution, it is **not unique** because non-invertibility implies there are dependent rows or columns, leading to multiple possible solutions.

### Answer:
- The correct option for when the system has a solution is **B. Sometimes**. This is because a non-invertible matrix may only sometimes yield solutions, depending on \( \mathbf{b} \).
  
### Is the solution unique?
- **No**, the solution is not unique for a non-invertible matrix, because there could be infinitely many solutions when the system is consistent (i.e., when \( \mathbf{b} \) is in the column space of \( A \)).

Would you like further details or clarifications on this? Here are some related questions:

1. What conditions make a matrix non-invertible?
2. How do we check if a system has infinitely many solutions?
3. Can a non-invertible matrix ever have a unique solution?
4. How does the rank of a matrix affect the solution of a system?
5. What is the difference between the null space and the column space of a matrix?

**Tip**: If you're working with non-invertible matrices, looking at the rank (number of independent rows/columns) can help determine the nature of the solutions.

When does the system A\mathbf{x} = \mathbf{b} have a solution for a non-invertible n \times n matrix A? Is the solution unique? Explain your reasoning.

Understand when the system A\mathbf{x} = \mathbf{b} has a solution for a non-invertible n \times n matrix. Explore the conditions for solution existence and uniqueness, using key concepts like column space and matrix rank.

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