I have this question:
Write the linear system
$$\begin{array}{rcr}-2x_1+x_2-4x_3 & = & 1 \\ x_1-2x_2 & = & -3 \\ x_1+x_2-4x_3 & = & 0 \end{array}$$
in the matrix-vector form $A\mathbf{x}=\mathbf{b}$.
Is this what they want?
$$
x_1*
\begin{bmatrix}
-2 \\ 1 \\ 1
\end{bmatrix}
+
x_2*
\begin{bmatrix}
1 \\ -2 \\ 1
\end{bmatrix}
+
x_3*
\begin{bmatrix}
-4 \\ 0 \\ -4
\end{bmatrix}
=
\begin{bmatrix}
1 \\ -3 \\ 0
\end{bmatrix}
$$
Best Answer
I guess the matrix-vector form here refers to the matrix A and the vector b. I would suggest to rewrite the equation in the following way
$$\begin{pmatrix}-2&1&-4\\1&-2&0\\1&1&-4\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}~=~\begin{pmatrix}1\\-3\\0\end{pmatrix}$$
To verify the L.H.S. you can just multiply the vector by the matrix and then your will get first guess.