How many multiplications must be performed to evaluate the d

How many multiplications must be performed to evaluate the determinant of an (n Times n) matrix according to the procedure of Problem 5 when n = 3, when n = 4, when n = 5, and for arbitrary n? (For n = 10, more than 3,000,000 multiplications are required. If a person were able to multiply two numbers and record the result at a rate of one per second, it would require 126 eight-hour days to find the determinant of a (10 Times 10) matrix using a cofactor expansion.

Solution

n =2

2x2 matrix 2 multiplications are required

3x3 matrix 2x3 = 6 multiplications are required

4x4 matrix 4x3x2 = 4!

Hence in general for nxn matrix n! multiplications are required.