Solutions of $Ax=b$ of square matrix $A$

Question

If A is a $5 \times 5$ matrix and the equation $Ax = b$ is consistent for every b in $R^5$; is it possible that for some $b$, the equation $Ax = b$ has more than one solution? Why or why not?

Answer 1

No. The matrix is invertible, and inverses are unique. So

$$Ax=b \iff x=A^{-1}b$$

is the unique vector $x$ satisfying that equality.

Answer 2

No. By the rank nullity theorem, the map $x \mapsto Ax$ is onto only if it is also one-to-one.