An Investigation into A Particular Operation on a Set of 100 Numbers - Essay Example

(Determinants and Cramer's Rule for Linear Equations, Undated) Both A and B are square matrices and thus the laws of determinants are applicable to them.Definition: There is no stand-alone definition of a determinant but it is defined in terms of a series of matrices as is evident hereafter.The determinant of an n x n matrix is defined as a sum of +/-1 times determinants of (n - 1) x (n - 1) matrices. (Determinants and Cramer's Rule for Linear Equations, Undated)Now that the technique of calculating determinants of matrices of any order has been somewhat explained a singular property of A is being touched upon.

A's Singularity Any 2 x 2 matrix derived out of any set of numbers in adjacent rows and columns of A has the same determinant.Example:Suppose,E = 78 7988 89Then, DetE = (78 x 89) - (79 x 88) = 6942 - 6952 = - 10This proves that all determinants of 2 x 2 matrices comprised of numbers in adjacent rows and columns are the same (-10) in the large 10 x 10 matrix A.This allows a general formula to be derived for the terms of all such matrices where N is a 2 x 2 matrix within A.

N = n n+1 n+10 n+11Here the numbers in the left diagonal - n and n+11 - have a periodicity of 11 while the numbers in the right diagonal - n+1 and n+10 - have a periodicity of 10. This is true of all 2 x 2 matrices comprised of numbers in adjacent rows and columns within A. And, detN = [n(n+11) - (n+1)(n+10)] = ( + 11n) - ( + 11n + 10) = - 10 This singularity gives rise to another that is mentioned in the calculations section and these two will make it easier to calculate the determinant of A.

For an n x n matrix if the 'i' order row is considered and the (n-1) x (n-1) matrix derived by crossing out the row and the column is also considered then the determinant of the original n x n matrix is as below. For the matrix A it is noted that all values of , where 'i' is the 1st row and the matrix is a 2 x 2 one, is -20. It is considered for the essay that is the difference and not the true determinant value. This assumption is now being checked out in the essay for any term. Thus, for calculating the value of the determinant of A it is also found that the value ofis alternately negative starting with the second term.

This makes it extremely easy to calculate the required value. Also, the operation has to be on a (n-1) x (n-1) matrix - a 9 x 9 matrix, as mentioned in the formula. This signifies that all numbers from 1-89 have to be considered. It is noticed on the right hand side of the equation that all even numbers are negative and all odd numbers are positive. Using the summation formula for series' in arithmetic progression - n/2[2a + (n - 1)d] - where n is the number of terms in the series, a is the first term and d is the common difference.

(Arithmetic Series, MathWorld, 2006) The singularities revealed by the investigation allows us to determine that any square matrix of any order n x n constituted of the particular sequence of numbers belonging to the set of all positive numbers starti