Linear Algebra: Does Ax=c Have a Unique Solution?

[VinnyCee]

Let A be a 4 X 4 matrix, and let \overrightarrow{b} and \overrightarrow{c} be two vectors in \mathbb{R}^4. We are told that the system A\,\overrightarrow{x}\,=\,\overrightarrow{b} has a unique solution. What can you say about the number of solutions of the system A\,\overrightarrow{x}\,=\,\overrightarrow{c}?

MY ANSWER:

The only way that the system A\,\overrightarrow{x}\,=\,\overrightarrow{c} has a unique solution is if A is invertable.

A is to be invertable.

A\,\overrightarrow{x}\,=\,\overrightarrow{c} has a unique solution.

But maybe the solution set \overrightarrow{c} does not have a unique solution because A is not truly invertable?

 

[Galileo]

If Ax=b has a unique solution, what does that tell you about the matrix A?