= Solution
For the <fourth-order two-stage Gauss collocation method>, $b_1=b_2=1/2$, $a_{11}=a_{22}=1/4$, $a_{12}=1/4-\sqrt3/6$ and $a_{21}=1/4+\sqrt3/6$. The weights are positive. For the matrix defining <algebraic stability>,
$$
m_{11}=m_{22}=2(1/2)(1/4)-(1/2)^2=0,
$$
and
$$
m_{12}=m_{21}=\frac12\left(\frac14-\frac{\sqrt3}6\right)
+\frac12\left(\frac14+\frac{\sqrt3}6\right)-\frac14=0.
$$
Hence \b[$M=0$ is positive semidefinite, so the method is algebraically stable]. Its two-stage contractivity estimate follows from part (b). The PDF has $a_{22}=1/4$; the extra $\sqrt3/6$ attached to that entry in the converted TeX is a transcription error.
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