matrices and elementary row operation. warm-up a= b=
TRANSCRIPT
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