Answer
See solution
Work Step by Step
\[
T(\mathbf{x})=\left[\begin{array}{c}
0 \\
x_{1}+x_{2} \\
x_{2}+x_{3} \\
x_{3}+x_{4}
\end{array}\right]=A\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right]
\]
To express $T(x)$ as $A x,$ write $T(x)$ and $x$ as column vectors, and then fill in the entries in $A$ by inspection
\[
T(\mathbf{x})=\left[\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1
\end{array}\right]\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right]
\]
Observe that since $T(x)$ and $x$ have four entries, $A$ must be a $4 \times 4$ matrix
