I need help for part b) only

Consider a product of m matrices A1312 - - - Am, (1) where A..- has size 115,1 x 11;. The total number of ops required to compute the result depends on the order in which we evaluate the matrix-matrix products. For example, for m = 4, we have the ve possibilities A1(A2[A3A4}) = A1((A2A3)A4) = [A1A2l(A3A4) = (A1(A2A3}}A4 = ((A1A2)A3)A4. In this problem we develop an efcient method for determining the optimal order for the matrix product in [1) and the number of ops if we use the optimal order. We denote by c113; the cost (optimal number of ops]I of computing A1A1+1 - . . Aj, where 1 g i '3' j 5 11].. Clearly 611 = [:22 = . .. = cmm = D, and [:12 = 2110111112, C23 = 211.111.2113., (We use the simplication [2g ljp-r a 230.11" for a product of a p x q and a g X 1' matrix.) We are interested in computng lm, the optimal number of ops for the entire product A1142 . . rAm. {a} Explain why ..= min c-+c 1+2 1-11111. CI; k=i112+1....1j1(%k k+1._'.| n: 1 k j} The minimum is over all 1.ralues of is that satisfy 1' S k: <. for example. ifi="1" and j="4," c14="min">