1. Matrices and Elementary Row Operation

2. Warm-up B= A=

3. 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.

4. 1
Type I
Type II
Type III
R3 => R1 1
(-⅓)R2 1
(-7)R3 + R1

5. Practice R2 => R1 (½)R1 (-2)R1 + R3 2 3 4 -1 -3 1 -1 2 3 4 -3 1 22 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

6. 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

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

8. Elementary Row Operationx +y +z = 6 2x -y 3 3x -z 1 6 2 -1 3
x=1, y=2, z=3

9. 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