What is Gaussian elimination?

1 Answer
Jul 6, 2018

Answer:

See below

Explanation:

Given: Gaussian elimination

Gaussian elimination, also known as row-reduction, is a technique used to solve systems of linear equations. The coefficients of the equations, including the constant are put in a matrix form.

Three types of operations are performed to create a matrix that has a diagonal of #1# and #0's# underneath:

#[ (1, a, b, c), (0, 1, d, e), (0, 0, 1, f) ]#

The three operations are:

  1. swap two rows
  2. Multiply a row by a nonzero constant (scalar)
  3. Multiply a row by a nonzero number and add to another row

Simple example. Solve for #x, y# using Gaussian Elimination:

#2x + 4y = -14#
#5x - 2y = 10#

Becomes:
#[ (2, 4, -14), (5, -2, 10) ]#

Multiply row 1 by #1/2#:
#[ (1, 2, -7), (5, -2, 10) ]#

Replace row 2 with: Multiply row 1 by #-5# and add to row 2:
#[ (1, 2, -7), (0, -12, 45) ]#

Divide row 2 by #-12#:
#[ (1, 2, -7), (0, 1, -15/4) ]# #=> x + 2y = -7; " "y = -15/4#

Use back substitution to solve for #x# and #y#:

#x + 2/1 (-15/4) = -7#

#x -30/4 = -7#

#x -15/2 = -14/2#

#x = -14/2 + 15/2 = 1/2#

Solution: #(1/2, -15/4)#