# How is this verified?

1. $Z=\frac{\hat{\beta}-0}{se(\hat{\beta})} \sim\ \mathcal{N}(0,1)$ where $H_0: \beta=0$
2. $\hat{\beta} \sim\ \mathcal{N}(\beta, (\sigma_u)^2(X'X)^{-1})$ where $\frac{plim(X'X)}{n} = \sigma_{xx}$

I reckon that in equation (2), the latter is the covariance matrix for the variable $X$, and the former is a kind of variance concept applied to matrices; i.e., the average of the square of a variable, since the square of the average is zero(?) hence not included? Is this correct reasoning?

The first equation is the distribution of a single parameter of $\hat{\beta}$, say for example $\hat{\beta}_j$, not for the entire parameter vector $\beta$.
The second equation is the distribution of the entire parameter vector $\hat{\beta}$. $\sigma_u^2$ is the variance of the residuals in the "true" model (can be estimated using a RSS-type formula), while $\sigma_{xx}$ is the covariance matrix of $X$ in the training set distribution (estimated by $X^{\prime}X / n$).
Notice that in simple linear regression, (2) reduces to (1). Both formulas depend on the assumption that the residuals $u$ are Gaussian.