Work: $\hat \beta\sim N(\beta, \sigma ^2 (X^TX)^{-1})$ Expanding, we have $u^2=(\hat \beta^T-\beta^T)\Gamma(\hat \beta - \beta)=\hat \beta ^T\Gamma\hat \beta-\hat \beta ^T\Gamma \beta - \beta ^T \Gamma \hat \beta+\beta ^T \Gamma \beta$, however this might not be the right approach. Which distribution would it be -- normal, $F$, $T$, $\chi^2$?
In previous part, if done correctly, $\Gamma ^{1/2}(\hat \beta - \beta)$ was shown to be normal with distribution $N(0, I_{p+1})$.