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Find the standard matrix A for T

[tex]

T\left(x_1,x_2,x_3\right)\,=\,\left(x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3\right)

[/tex]

[tex]

\mathbf{v}\,=\,\begin{bmatrix}

x_1\\

x_2\\

x_3

\end{bmatrix}\,\,\,\,T\left(\mathbf{v}\right)\,=\,T\,\begin{bmatrix}

x_1\\

x_2\\

x_3

\end{bmatrix}\,=\,\begin{bmatrix}

x_1 + x_2 + x_3\\

x_2 + x_3\\

3x_1 + x_2\\

2x_2 + x_3

\end{bmatrix}\,=\,\begin{bmatrix}

1 & 1 & 1\\

0 & 1 & 1\\

3 & 1 & 0\\

0 & 2 & 1

\end{bmatrix}

[/tex]

Is that correct? I feel like I'm just walking down a blind alley with this problem. The text is sorta convoluted in this section and I can't seem to find any supplementary material that I feel is helpful.

Thanks.