# Matrices and Elementary Row Operation

## Presentation on theme: "Matrices and Elementary Row Operation"— Presentation transcript:

Matrices and Elementary Row Operation

Warm-up B= A=

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.

1 Type I Type II Type III R3 => R1 1 (-⅓)R2 1 (-7)R3 + R1

Practice R2 => R1 (½)R1 (-2)R1 + R3 2 3 4 -1 -3 1 -1 2 3 4 -3 1 2
2 3 4 -1 -3 1 -1 2 3 4 -3 1 R2 => R1 2 -4 6 -2 1 3 -3 5 (½)R1 1 -2 3 -1 -3 5 2 1 2 -4 3 -2 -1 5 1 2 -4 3 -2 -1 -3 13 -8 (-2)R1 + R3

1 -2 3 9 5 2 -5 17 1. -2R1+R3 2. R2+R3 3. (½)R3 AUGMENTED MATRIX 1 -2
-2y +3z = 9 y 5 2x -5y +5z 17 1 -2 3 9 5 2 -5 17 1. -2R1+R3 1 -2 3 9 5 -1 2. R2+R3 AUGMENTED MATRIX 1 -2 3 9 5 2 -5 17 1 -2 3 9 5 2 4 3. (½)R3 1 -2 3 9 5 2

Using back-substitution
SOLVE FOR X,Y,Z Using back-substitution 1 -2 3 9 5 2 x=1, y=-1, z=2

Elementary Row Operation
x +y +z = 6 2x -y 3 3x -z 1 6 2 -1 3 x=1, y=2, z=3

x=8, y=10, z=6 x=-4, y=-3, z=6 2x -y +3z = 24 -z 14 7x -5y 6 x +2y -3z
-28 4y +2z -x +y -z -5 x=-4, y=-3, z=6

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