Matrices and Elementary Row Operation. Warm-up A= B=

Matrices Matrices and and

Elementary Row Elementary Row OperationOperation

Warm-up

A=A= B=B=

Elementary Row Operations

•A SQUARE matrix is an elementary matrix if it is obtained from the identity matrix by a single elementary row operation.

• Elementary row operations:

Type I: Interchange two rows.

Type II: Multiply a row by a non-zero constant.

Type III: Add a multiple of one row to another.

Type I

Type II

Type III

1 0 00 1 00 0 1

1 0 00 1 00 0 1

1 0 00 1 00 0 1

R3 => R1

(-⅓)R2

(-7)R3 + R1

Practice0 2 3 4

-1 2 0 3

2 -3 4 1

2 -4 6 -2

1 3 -3 0

5 -2 1 2

1 2 -4 3

0 3 -2 -1

2 1 5 -2

R2 => R1

(½)R1

(-2)R1 + R3

-1 2 0 3

0 2 3 4

2 -3 4 1

1 -2 3 -1

1 3 -3 0

5 -2 1 2

1 2 -4 3

0 3 -2 -1

0 -3 13 -8

x-

2y+3z

= 9

y+3z

= 5

2x

-5y

+5z

= 17

1. -2R1+R3

2. R2+R3

3. (½)R3

1-2

3 9

0 1 3 5

2-5

517

AUGMENTED MATRIX

1 -2 3 90 1 3 5

2 -5 517

1 -2 3 90 1 3 50 -1 -1 -1

1 -2 3 90 1 3 50 0 2 4

1 -2 3 9

0 1 3 5

0 0 1 2

SOLVE FOR X,Y,Z

1 -2 3 9

0 1 3 5

0 0 1 2

x=1, y=-1, z=2

Using back-substitution

Elementary Row Operation

x +y +z = 62x

-y +z = 3

3x

-z = 0

1 1 1 62 -1 1 33 0 -1 0

x=1, y=2, z=3

2x

-y+3z

= 24

2y -z = 147x

-5y

= 6

x+2y

-3z =-

28

4y+2z

= 0

-x +y -z = -5

x=8, y=10, z=6

x=-4, y=-3, z=6