Section 8.6

Elimination using Matrices

Matrix Method

The method computers use. The equations need to be in standard form. The coefficients and constants are translated

into a rectangle array. Make the rectangle array into row echelon form. Find answer by using back substitution.

Standard Form

Write the linear equation in the form Ax + By = C

If variables are different, go in alphabet order. A, B and C do not have any restrictions, but life

is easier if they are integers {.., -2, -1, 0, 1, 2, ..}

Standard Form to Rectangle Array

{Ax +By=CDx +Ey=F}Write the standard form of system of linear equations

In rectangle array form

[A B CD E F]

Example

Write the system of linear equations in standard form and in rectangle array

Standard form

Rectangle array

{ y=x+23 y−2 x=4}

{1 y−1 x=23 y−2 x=4}

[1 −1 23 −2 4]

Row Echelon Form

The values on the diagonal need to be ones. The values below the diagonal need to be

zeros. The other values can be any number, using

lower class letters because they could change values.

[1 b c0 1 f ]

Back Substitution

Given the row echelon form.

Rewrite it in standard form.

Solve the bottom equation then the top equation y = f x + by = c x + bf = c x = c - bf

[1 b c0 1 f ]

{1 x+by=c0 x+1 y= f }

Example

[1 2 00 1 −1]

Given the matrix find the values for x and y

1. Write the bottom row as an equation0x + 0y = -1

2. Solve y = -13. Write the top row as an equation

1x + 3y = 04. Substitute the answer we found for y

1X + 3(-1) = 05. Solve X = 36. Write answer in ordered form (3, -1)

Rules to make the Row Echelon Form

The following operations produces a row equivalent matrix

1. Interchanging any two rows.2. Multiplying all elements of a row by a nonzero

constant.3. Adding two rows together.

You can blend rules together, especially 2 and 3Each step needs to include the proper rule.

Interchanging two rows

r i⇔ r jThe proper format for this rule isi and j are the specific rows you will swap

Example. Write the matrix in row echelon form.

[0 1 −11 2 −5]

Multiplying a row by a constant

The proper format for this rule is i is the row c is the constant

The constant will be multiplied to all values in the row.

Example: Write the matrix in row echelon form.

cr i→ ri

[−2 4 20 1 −1]

Adding two rows together

The proper format for this rule is i and j are the two rows to be added i is the row you will be placing the answer into

Example Write the matrix in row echelon form

r i+r j→ r i

[ 1 −2 1−1 3 3]

Example

Solve the system of linear equations.

{ y=x+23 y−2 x=4}

Example

Solve the system of linear equations

{3 x−y=19 x−3 y=1}

No Solution and Infinite Solutions

If the matrix looks like then you have a no solution.

If the matrix looks like then you have an infinite solutions.

[1 x y0 0 y≠0]

[1 x y0 0 0]

Example

Solve the system of linear equations

{4 x−2 y=62 x−3= y }

Homework

Section 8.7 7, 8, 9, 10, 11, 12