Is it possible for an augmented matrix of a $4 \times 4$ system to have infinitely many solutions without having a row of zeroes? I think it's not but not too sure.
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$\begingroup$To be more precise, consider the system $Ax=b$ where $A \in \mathbb{R}^{4 \times 4}$ consists of all ones and $b\in \mathbb{R}^{4 \times 1}$ consists of all ones as well. In that case, we have infinite number of solutions though there is no zero rows.
However, if the augmented matrix that you intended to ask is already in REF form, then without zero rows, every column is a pivot column and there is no free variable. Hence the solution is unique.
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