Multiply a list of matrices by a list of vectors

MapThread[Dot, {A, x}]   // TeXForm

$\left( \begin{array}{cc} 1 & 1 \\ 4 & -4 \\ \end{array} \right)$

Perhaps just:

Dot @@@ Transpose[{A, x}]

Following Belasarius in a somewhat more transparent fashion, you Apply[ ] the Dot function to A and x. The Transpose is used to get the A and x into a single list and the "1" causes the Apply to to work at the correct level.

 Apply[Dot, Transpose[{A, x}], 1]