Does the augmented matrix have no solution??

Question

I'm preparing for an exam and one of the sample papers had this question.

$$ \left[\begin{array}{rrrr|r} 1 & 2 & 2 & 4 & 8 \\ 1 & 2 & 2 & 0 & 8 \\ 0 & 1 & 1 & 1 & 3 \\ 0 & 2 & 2 & 2 &6 \end{array}\right] $$

Since $R_4=2\times R_3$ then that means that the matrix has no solution right??

Answer 1

Hint:- For the system $Ax=b$, where $A$ is $m×n$, $x$ is $n×1$, $b$ is $m×1$, the augmented matrix is $C=[A:b]$ and it has

1. a unique solution if $rank(A)=rank(C)=n$

2. infinitely many solutions if $rank(A)=rank(C)<n$

3. no solution if $rank(A)\ne rank(C)$