$$\text{Let} \;
A =
\begin{bmatrix}
1 & 2 & 3\\
2 & 5 & 3\\
1 & 0 & 8\\
\end{bmatrix},\;
D =
\begin{bmatrix}
0 & 0 & 3\\
3 & 1 & 1\\
0 & 2 & 0\\
\end{bmatrix},\;
\text{and b =}
\begin{bmatrix}
21\\
9\\
-1\\
\end{bmatrix} $$
Determine whether the systems $A$x = b and $D$x = b are consistent or inconsistent. If consistent, solve the system.
I setup an augmented matrix [$A$|b] and found
$$\text{x =}
\begin{bmatrix}
-705\\
231\\
88\\
\end{bmatrix} $$
However, using this vector x, $D$x $\ne$ b.
Does this mean the system is inconsistent?
Best Answer
No. My response would be why do you think $D \mathbf x = \mathbf b$ would hold for the same $\mathbf x$ that makes $A\mathbf x=\mathbf b$ true?
There are really two systems here: $A\mathbf x = \mathbf b$ and $D\mathbf x = \mathbf b$. The consistency of one has nothing to do with the consistency of the other.
A system is consistent if and only if it has at least one solution. So if you found a solution for the system $A\mathbf x = \mathbf b$, then the system is consistent. And you did find the right solution. Now turn your attention towards $D \mathbf x = \mathbf b$, keeping it completely separate from $A\mathbf x = \mathbf b$.