Solution
= Solution
At the maximum-likelihood estimates,
$$
\widehat\sigma_1^2=\frac1n\|Y-X\widehat\beta\|_2^2,
$$
so
$$
-2\ell(\widehat\beta,\widehat\sigma_1^2)
=n\{\log(2\pi\widehat\sigma_1^2)+1\}.
$$
There are $p$ regression coefficients and one variance parameter. Adding twice this parameter count gives
$$
\operatorname{AIC}(M_1)
=n\{\log(2\pi\widehat\sigma_1^2)+1\}+2(p+1).
$$