Question:

Consider the matrices A1, A2, A3 and A4, of sizes: 2 15, 15 10, 10 1 and 1 20, respectively. 

(1)     If B is a matrix of size 10 15, can the following matrix multiplications be performed, and if so, what will be the size of the matrix obtained as a result of the multiplication?
A2 B  and  B A3.
(2)     Using dynamic programming approach, find the minimum number of multiplications necessary to compute the product of matrices:
A1 ´ A2 ´ A3 ´ A4.
Indicate the corresponding optimal position of brackets.

Answer:

(1)

A2B = 15*15

BA3: Error.

(2)

# -*- coding: utf-8 -*-

class Matrix:
  def __init__(self, row_num=0, col_num=0, matrix=None):
    if matrix != None:
      self.row_num = len(matrix)
      self.col_num = len(matrix[0])
    else:
      self.row_num = row_num
      self.col_num = col_num
    self.matrix = matrix


def matrix_chain(matrixs):
  matrix_num = len(matrixs)
  count = [[0 for j in range(matrix_num)] for i in range(matrix_num)]
  flag = [[0 for j in range(matrix_num)] for i in range(matrix_num)]
  for interval in range(1, matrix_num + 1):
    for i in range(matrix_num - interval):
      j = i + interval
      count[i][j] = count[i][i] + count[i + 1][j] + matrixs[i].row_num * matrixs[i + 1].row_num * matrixs[j].col_num
      flag[i][j] = i
      for k in range(i + 1, j):
        temp = count[i][k] + count[k + 1][j] + matrixs[i].row_num * matrixs[k + 1].row_num * matrixs[j].col_num
        if temp < count[i][j]:
          count[i][j] = temp
          flag[i][j] = k
  traceback(0, matrix_num - 1, flag)
  return count[0][matrix_num - 1]


def traceback(i, j, flag):
  if i == j:
    return
  if j - i > 1:
    print(str(i + 1) + '~' + str(j + 1), end=': ')
    print(str(i + 1) + ":" + str(flag[i][j] + 1), end=',')
    print(str(flag[i][j] + 2) + ":" + str(j + 1))
  traceback(i, flag[i][j], flag)
  traceback(flag[i][j] + 1, j, flag)


matrixs = [Matrix(2, 15), Matrix(15, 10), Matrix(10, 1), Matrix(1, 20)]
result = matrix_chain(matrixs)
print(result)

The result is 220

Reference: https://www.jb51.net/article/129168.htm