Need help in solving a linear algebra ( System of Equations) quiz problem I am solving previous year quiz problem of my class and I am unable to solve this question in linear algebra.

Let $A \in M_{m \times n}(\Bbb{R})$ and let $b_0 \in \Bbb{R}^m$. Suppose the system of equations $Ax = b_0$ has a unique solution. Which of the following statement(s) is/are true?

*

*$Ax = b$ has a solution for every $b \in \Bbb{R}^m$.

*If $Ax = b$ has a solution then it is unique.

*$Ax = 0$ has a unique solution.

*$A$ has rank $m$.


( Multiple answers can be correct)
Answer is :

 2, 3

I know  the theorem that if $A$ is invertible then $Ax=0$ has only trivial solution and $A$ has rank $m$ and then $Ax=b$ has a solution for every $b \in \mathbb{R}^m$. But the problem is, how I can be sure if $A$ is invertible?
Kindly help.

Recall that the system $Ax=b$ is solvable iff $b$ is an element of the span of the column vectors of $A$.  Now if this solution is unique the column vectors of $A$ are linearly independent.  Does this clear up things?
