= Solution
The tableau is the three-stage <Lobatto IIIA method>. To check its <order of a Runge-Kutta method> directly, write $e=(1,1,1)^T$, $c=Ae=(0,1/2,1)^T$, and $C=\operatorname{diag}(c)$. The eight <fourth-order conditions for a Runge-Kutta method> evaluate to
$$
\begin{gathered}
b^Te=1,\qquad b^Tc=\frac12,\qquad b^Tc^2=\frac13,\qquad b^TAc=\frac16,\\
b^Tc^3=\frac14,\qquad b^TCAc=\frac18,\qquad
b^TAc^2=\frac1{12},\qquad b^TA^2c=\frac1{24}.
\end{gathered}
$$
Powers of $c$ here mean componentwise powers. For example, $Ac=(0,1/8,1/2)^T$, $Ac^2=(0,1/24,1/3)^T$, and $A^2c=(0,1/48,1/6)^T$, which make the last three checks immediate. These <Butcher order conditions> establish order at least four for general nonlinear <ordinary differential equations>.
It is not order five: part (b)'s <stability function> has expansion
$$
R(z)=1+z+\frac{z^2}2+\frac{z^3}6+\frac{z^4}{24}+\frac{z^5}{144}+O(z^6)
=e^z-\frac{z^5}{720}+O(z^6).
$$
The fifth coefficient already fails on the <Dahlquist test equation>. \b[The order is exactly four.]
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