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 … M_n can be done by parenthesizing in different ways. Define M_iM_i+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 A_5X40, A_40×6, A_6×20, A_20×5
According to matrix chain multiplication:

Therefore, selected pairs are:

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

So, option (A) is correct.

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