Question

Show that the elementary row operations do not affect the solution set of a linear system.

Answer 1

@ganeshie8 @jim_thompson5910

Answer 2

this may be of help

https://www.math.ucdavis.edu/~linear/old/notes3.pdf

Answer 3

1) Exchanging rows is same as changing the order of given equations - it wont affect the solutions
2) Multiplying a row by a non zero constant is same as multiplying a nonzero constant to both sides of an equation - it wont affect the solution

Answer 4

3) Adding a multiple of a row to another row :
\[\large (a_{j1} + ta_{k1})x_1 + (a_{j2} + ta_{k2})x_2 + \cdots + (a_{jn} + ta_{kn})x_n \]

\[\large (a_{j1}x_1 + a_{j2}x_2 + \cdots + a_{jn}x_n)+ (a_{k1}x_1 + a_{k2}x_2 + \cdots + a_{jn}x_n) \]

Clearly if a solution satisfies both the equations, it satisfies above combined equation also