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**Matrices and Elementary Row Operation**

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Warm-up B= A=

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**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.

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

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**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

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**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

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**Using back-substitution**

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

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**Elementary Row Operation**

x +y +z = 6 2x -y 3 3x -z 1 6 2 -1 3 x=1, y=2, z=3

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**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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