= Solution
For the <Dahlquist test equation>, put $z=h\lambda$. Eliminating the stages of the <Runge-Kutta method> gives the <stability function>
$$
R(z)=1+zb^T(I-zA)^{-1}e
=\boxed{\frac{1+z/3}{1-2z/3+z^2/6}}.
$$
Its poles are $2\pm i\sqrt2$, strictly in the right half-plane. To prove <A-stability> directly, write $z=-r+iy$ with $r\geq0$. With $D(z)=1-2z/3+z^2/6$ and $N(z)=1+z/3$, elementary expansion gives
$$
|D(z)|^2-|N(z)|^2
=2r+\frac23r^2+\frac29r^3+\frac1{36}r^4
+\frac29ry^2+\frac1{18}r^2y^2+\frac1{36}y^4\geq0.
$$
Therefore $|R(z)|\leq1$ throughout the closed left half-plane; the inequality is strict except at $z=0$. \b[The method is <A-stable>.] Moreover $R(z)\to0$ as $|z|\to\infty$, so the <Radau IIA method> is also <L-stable>.
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