# constant matrix times Multivariate Gaussian distribution?

Suppose I have a multivariate Gaussian distribution x and a constant matrix A. I know how to calculate the mean and covariance of Ax but how can I prove that Ax will also be multivariate gaussian??

In addition to characteristic functions, one can use Jacobians to find an expression for RV transformations, i.e. when we set $$\mathbf{Y}=\mathbf{AX}+\mathbf{b}$$, it's further simplified to: $$f_\mathbf{Y}(\mathbf{y})=\frac{1}{\vert\mathbf{A}\vert}f_\mathbf{X}(\mathbf{A}^{-1}\vert\mathbf{y}-\mathbf{b}\vert)$$
Since $$\vert\mathbf{A}\vert$$ is constant, $$f_{\mathbf{Y}}(y)\propto f_\mathbf{X}(\mathbf{A}^{-1}\vert\mathbf{y}-\mathbf{b}\vert)$$, and since $$f_\mathbf{X}(\mathbf{.})$$ is in MV normal form, so is $$f_\mathbf{Y}(\mathbf{y})$$.