Matrix "divided" by a matrix. Can this be done?

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So I am working on a problem and which basically looks like this

$- \frac{x^t A^{-1}x}{M x^TA^{-2}x}$ where $M$ is a scalar, $x$ is a vector and $A$ is a matrix (which in the actual problem is the hessian matrix and so is positive definite).

My question is, can the vector matrix products essentially "cancel" to leave me with just $\frac{1}{MA^{-1}}$?