# 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. $((A_1A_2)A_3)A_4$