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.