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?