Let's represent multivariate normal distribution as an exponential family:
\begin{align} f_X( x | \Theta ) = h(x)g(\Theta)\exp( \eta(\Theta) \cdot T(x) ) \end{align}
Where natural parameters:
\begin{align} \eta(\Theta) = \left[ \begin{matrix} \Sigma^{-1} \mu \\ -\frac12 \Sigma^{-1} \end{matrix} \right] \end{align}
Sufficient statistic:
\begin{align} T(x) = \left[ \begin{matrix} x \\ xx^T \end{matrix} \right] \end{align}
Does $\eta(\Theta) \cdot T(x)$ denote a dot product?
If so then we can't get a scalar value. We need some more complex function then dot product - for instance trace: $\operatorname*{Tr}(\Sigma^{-1} xx^T) = \operatorname*{Tr}(x^T\Sigma^{-1} x)= x^T\Sigma^{-1} x$
Update
What are the proper identical transformations to get:
\begin{align} \eta(\Theta) \cdot T(x) = \\ \left[ \begin{matrix} \Sigma^{-1} \mu \\ -\frac12 \Sigma^{-1} \end{matrix} \right] \cdot \left[ \begin{matrix} x \\ xx^T \end{matrix} \right] = \\ ??? = \\ -\frac12 x^T \Sigma^{-1} x + \mu^T \Sigma^{-1} x \end{align}
However, the inner product has an alternative form $x^T\Sigma x=\sum_{i,j} x_i x_j \sigma_{ij}$ where $\Sigma=\{\sigma_{ij}\}$.