Solve a system of linear equations

By reducing a matrix

Pamela Leutwyler

0433

02

0322

zyxw

zyxw

zyxw

Describe the solutions to:

0433

02

0322

zyxw

zyxw

zyxw

Describe the solutions to:

01433

01211

01322

The coefficient matrix:

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

01322

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

01322

Replace row 1 with row 1 – row 2

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

01322

Replace row 1 with row 1 – row 2

-1

1 1 1 0

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

00111

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

00111

Replace row 2 with row 2 – row 1

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

00111

Replace row 2 with row 2 – row 1

-1

0433

02

0322

zyxw

zyxw

zyxw

01433

01211

00111

Replace row 2 with row 2 – row 1

-10 0 1 1

0433

02

0322

zyxw

zyxw

zyxw

01433

01100

00111

0433

02

0322

zyxw

zyxw

zyxw

01433

01100

00111

Replace row 3 with row 3 – 3 row 1

0433

02

0322

zyxw

zyxw

zyxw

01433

01100

00111

Replace row 3 with row 3 – 3 row 1

-3

0433

02

0322

zyxw

zyxw

zyxw

01433

01100

00111

Replace row 3 with row 3 – 3 row 1

-3

0 0 1 1

0433

02

0322

zyxw

zyxw

zyxw

01100

01100

00111

0433

02

0322

zyxw

zyxw

zyxw

01100

01100

00111

Replace row 3 with row 3 – row 2

0433

02

0322

zyxw

zyxw

zyxw

01100

01100

00111

Replace row 3 with row 3 – row 2

-1

0433

02

0322

zyxw

zyxw

zyxw

01100

01100

00111

Replace row 3 with row 3 – row 2

-1

0 0

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

00111

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

00111

Replace row 1 with row 1 – row 2

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

00111

Replace row 1 with row 1 – row 2

-1

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

00111

Replace row 1 with row 1 – row 2

-1

0 -1

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

01011

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

01011

The matrix is in REDUCED ECHELON FORM

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

01011

The matrix is in REDUCED ECHELON FORM

11

Every FNZE is a 1First NonZero Entry in a row of a matrix

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

01011

The matrix is in REDUCED ECHELON FORM

11

Every FNZE is a 1Every FNZE is to the right of the FNZE above it.

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

01011

The matrix is in REDUCED ECHELON FORM

11

Every FNZE is a 1Every FNZE is to the right of the FNZE above it.

00

0

0

In a column containing a FNZE, every other element is a 0

0433

02

0322

zyxw

zyxw

zyxw

00000

01100

01011

Interpret :

0433

02

0322

zyxw

zyxw

zyxw

Interpret :

00000

01100

01011

zyxw

zyxw

zyxw w= -x +z

0433

02

0322

zyxw

zyxw

zyxw

Interpret :

00000

01100

01011

zyxw

zyxw

zyxw w= -x +z

y= -z

0433

02

0322

zyxw

zyxw

zyxw

Interpret :

w= -x +z

y= -z

0433

02

0322

zyxw

zyxw

zyxw

Interpret :

w= -x +z

y= -zx= x

z= z

x and z are theindependent variables

0433

02

0322

zyxw

zyxw

zyxw

Interpret :

w= -x +z

y= -zx= x

z= z zz

zy

xx

zxw

1

1

1

11

0433

02

0322

zyxw

zyxw

zyxw

Interpret :

zz

zy

xx

zxw

1

1

1

11

1

1

0

1

0

0

1

1

zx

z

y

x

w

Every solution is a linear combination of these vectors