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 5x40,40×6,6x20 and 20×5 respectively. In the parenthesization of A₁A₂A₃A₄ that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are