2025-02-Algorithm/reviews/R8d.md

Review 8-4

• Hajin Ju, 2024062806

Problem 1

What is the dimension of the matrix product AB if A is a p\times q matrix and B is a q\times r matrix?

Solution 1

p\times r

Problem 2

Count the number of scalar multiplications to multiply A and B where A is a p\times q matrix and B is a q\times r matrix.

Solution 2

pqr

Problem 3

Count the number of scalar multiplications where the dimensions of A_1, A_2, and A_3 are 10\times 100, 100\times 5 and 5\times 50, respectively.

1. (A_1A_2)A_3
2. A_1(A_2A_3)

Solution 3

1. 5000 + 2500 = 7500
2. 25000 + 50000 = 75000

Problem 4

Fully parenthesize the product A_1A_2A_3A_4. (There are five distinct ways.)

Solution 4

1. (A_1A_2)(A_3A_4)
2. (A_1(A_2A_3))A_4
3. A_1(A_2(A_3A_4))
4. A_1((A_2A_3)A_4)
5. ((A_1A_2)A_3)A_4