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what is full parenthesization?? when we do it and need of it??

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since A1  has dimension  m*n
               A2  has dimension  p*q

               A3  has dimension  q*r

          A4  has dimension  r*s

      A5 has dimension  m*n

(A1 A2 A3 A4 A5)m*s     so   it is equal to = 
 (   (   (  ( A1*A2 )*A3 ) *A4 )*A5  )

so here we found that we need 4 pair of parenthesis for full pair of parenthesization.

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parenthesis for full pair of parenthesization.

What is the mean of this? 

this is the only way to solve multiplication for 5 variables? suppose I assume some random size for matrix and perform multiplications. Does it give the minimum number of multiplication?

what are the different types of orders are possible for multiplications?

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