Question

What is Gaussian elimination?

Answer 1

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)`