Show the validity of Wald's test in GLM for testing contrasts. Let $A$ be a $q \times p$ matrix, $q \leq p$, of rank $\mathrm{q}, \zeta$ a $q$-dimensional column vector and $\beta$ a $p$-dimensional column vector. If $\tilde{\beta} \sim N_{p}\left(\beta,\left(X^{T} W X ight)^{-1} ight)$, prove that, under $H_{0}: A \beta=\zeta$, $$ (A \tilde{\beta}-\zeta)^{T} \Sigma^{-1}(A \tilde{\beta}-\zeta) \sim \chi_{q}^{2} $$ where $\Sigma=A\left(X^{T} W X ight)^{-1} A^{T}$. All inverses are assume exit