GAUSSIAN ELIMINATION

Carl Friedrich Gauss

Carl Friedrich Gauss is sometimes referred to as the "Prince of Mathematicians" and the "greatest mathematician since antiquity". He has had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.
Gauss was a child prodigy. There are many anecdotes concerning his precocity as a child, and he made his first ground-breaking mathematical discoveries while still a teenager.
At just three years old, he corrected an error in his father payroll calculations, and he was looking after his father’s accounts on a regular basis by the age of 5. At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually 50 pairs of numbers, with each pair summing to 101, total 5,050). By the age of 12, he was already attending gymnasium and criticizing Euclid’s geometry.
Although his family was poor and working class, Gauss' intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at 15, and then to the prestigious University of Göttingen (which he attended from 1795 to 1798). It was as a teenager attending university that Gauss discovered (or independently rediscovered) several important theorems.

Graphs of the density of prime numbers

At 15, Gauss was the first to find any kind of a pattern in the occurrence of prime numbers, a problem which had exercised the minds of the best mathematicians since ancient times. Although the occurrence of prime numbers appeared to be almost competely random, Gauss approached the problem from a different angle by graphing the incidence of primes as the numbers increased. He noticed a rough pattern or trend: as the numbers increased by 10, the probability of prime numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4 chance of getting a prime in the number from 1 to 100, a 1 in 6 chance of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to 10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware that his method merely yielded an approximation and, as he could not definitively prove his findings, and kept them secret until much later in life.


Gaussian Elimination
                     It was introduce as a procedure for solving a system of linear equations.
Matrix 
        - is a rectangular array of numbers a, symbol or expression arranged  in rows and columns

Entry
        - Is the individual items in the matrix 
                    aij
Row Subscript
         -  The index i above is called row subscript because it identifies the row in which the entry lies

Column Subscript
         - The index j is called the Column Subscript because it identifies the column in which the entry lies

Square of Order n
        - A matrix with m rows and n columns (an m x n ) is said to be the size. If m=n then the matrix is called square of order n


Main Diagonal Entries
              a11, a12, a33...
Augmented Matrix
     -the matrix derived from the coefficient and constant terms of a system of linear equation
Coefficient Matrix
     - The matrix containing only the coefficients of the system
Elementary Row Operations
1. Interchange two equations
2. Multiply an equation by a non-zero constant
3.Add a multiple of an equation to another equation
 Examples:





















references : http://www.storyofmathematics.com/19th_gauss.html
                   http://www.math.dartmouth.edu/archive/m23s06/public_html/handouts/row_reduction_examples.pdf