Linear Models — consistency of the normal equations

For a linear model, $\mathbf{Y} = \mathbf{X}\beta + \varepsilon$, $\mathbb{E}(\varepsilon) = 0$, $D(\varepsilon) = \sigma^2\mathbf{I}$ where $D$ stands for variance-covariance matrix and $\mathbf{I}$, for the identity matrix of order $n$, I. The observational equation $\mathbf{Y} = \mathbf{X}\beta$ is always consistent. II. The normal equation $\mathbf{X'X}\beta = \mathbf{X'Y}$ always admits a solution. Which of the statements given above is/are correct?