GATE | Sudo GATE 2020 Mock I (27 December 2019) | Question 55

Assume that multiplying a matrix M1 of dimension a×b with another matrix M2 of dimension bxc requires abc scalar multiplications. Computing the product of n matrices M1M1M3 … Mn can be done by parenthesizing in different ways. Define MiMi+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain M1M2M3M4M5M6 using parenthesization (M1(M2M3))(M4(M5M6)), M2M3 and M5M6 are only explicitly computed pairs.

Consider a matrix multiplication chain A1A2A3A4, where matrices A1, A2, A3 and A4 are of dimensions 5×40,40×6,6×20 and 20×5 respectively. In the parenthesization of A1A2A3A4 that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are
(A) A1A2 and A3A4
(B) A2A3 only
(C) A3A4 only
(D) A1A2 and A2A3

Answer: (A)

Explanation: Given matrix A5X40, A40×6, A6×20, A20×5
According to matrix chain multiplication:

Therefore, selected pairs are:

[A3, A4] = 600
[A1, A2] = 1200 

So, option (A) is correct.

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