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I would like the number $a$ to be taken out:

$Assumptions = (a) \[Element] Complexes && (X) \[Element] 
    Matrices[{n, n}];

(a X).(a IdentityMatrix[n]) // FullSimplify 

Is there a way to make this product to yield $a^2 X$ instead of (X a).(a IdentityMatrix[n])? And to simplify the IdentityMatrix[n] as well?

P.S. Mathematica 10.

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Use TensorExpand[] instead:

Assuming[a ∈ Complexes && X ∈ Matrices[{n, n}], 
         TensorExpand[(a X).(a IdentityMatrix[n])]]
   a^2 X
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