= Solution
Let $RSS_0$ and $RSS_1$ be the residual sums of squares under the restricted model with $p_0$ coefficients and the full model with $p$ coefficients. Assuming $X$ has full column rank, the <nested-model F-test> uses
$$
\boxed{F=\frac{(RSS_0-RSS_1)/(p-p_0)}{RSS_1/(n-p)}.}
$$
Under the null hypothesis this has the $F_{p-p_0,n-p}$ distribution.
Maximizing the Gaussian likelihood over $\beta$ and the unknown variance gives $\widehat\sigma^2=RSS/n$. Thus the <likelihood-ratio test statistic> is
$$
-2\log\Lambda=n\log\frac{RSS_0}{RSS_1}.
$$
Since
$$
\frac{RSS_0}{RSS_1}=1+\frac{p-p_0}{n-p}F,
$$
we obtain
$$
\boxed{-2\log\Lambda
=n\log\left(1+\frac{p-p_0}{n-p}F\right).}
$$
The logarithm is strictly increasing, so the two test statistics give exactly the same rejection ordering.
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