Relationships between m, n, r and Possible Solutions

$m\times n$ matrix $A$ of rank $r$.

 
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Cases

$r=n$

Here A is invertible. $N(A)=Z$. Either 0 or 1 particular solution $x\_p$ exists. $R=\left[\begin{array}{c}I0\end{array}\right]$

$r=m$

$Ax=b$ is solvable for every right-hand side. $R=\left[\begin{array}{cc}I& F\end{array}\right]$

$r=m=n$

Invertible square matrix. It reduces to $R=I$ and therefore has only one solution for any $Ax=b$. $N(A)=Z$