= Solution
Apply the <Runge-Kutta method> to the <Dahlquist test equation> and put $z=h\lambda$. Its <stability function> is
$$
R(z)=1+z\,b^T(I-zA)^{-1}\mathbf1=\frac{P(z)}{Q(z)},\qquad Q(z)=\det(I-zA).
$$
Since $A$ is an <invertible matrix>, $Q$ has <polynomial degree> $s$. The <adjugate matrix> formula gives $\deg P\le s$, but the additional condition gives
$$
\lim_{z\to\infty}R(z)=1-b^TA^{-1}\mathbf1=0,
$$
so actually $\deg P\le s-1$. The specified <order of a numerical method> implies $R(z)=e^z+O(z^{2s})$ near zero. Therefore the <Padé identification of a Radau stability function> makes $R$ the $[s-1/s]$ <Padé approximant> to the <exponential function>: its numerator and denominator have the required degrees, and its <Taylor series> agrees through degree $2s-1$.
In the permitted <A-stability of near-diagonal exponential Padé approximants>, take $m=s-1$, $n=s$. Then $n-2\le m\le n$, so \b[the scheme is <A-stable>]. The same <stability function> also tends to zero at infinity, giving <L-stability>.
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