Linear Transformation of a Multivariate Normal via MGF

Let $\mathbf{X} \sim N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ be a $p$-dimensional normal random vector, and let $\mathbf{A}$ be a fixed $m \times p$ matrix. Using the moment-generating function, prove that $\mathbf{Z} = \mathbf{A}\mathbf{X}$ is multivariate normal and determine its mean vector and covariance matrix.