simplify an expression of vectors an matrices

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I want to simplify this term where:

  1. s is a n x 1 vector
  2. z and z' are a n x 1 vector
  3. and D is a n x n matrix

I know that by changing the place of a matrix inside the multiplication will mess up the result. For example, can i multiply by s the whole equation so the 1/s will go away? or use z as a common factor? essentially what rules can i follow in these type of equations that they won't break them?

$$ \frac{1}{s} * z' = \frac{1}{s} * z - \frac{1}{1+ tr(D*\frac{1}{s}*z)}*\frac{1}{s}*z*D*\frac{1}{s}*z $$