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# Math 125 Week 3 Matrices REF/RREF MATH 125

UIUC

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This 5 page Class Notes was uploaded by Jason Notetaker on Wednesday September 7, 2016. The Class Notes belongs to MATH 125 at University of Illinois at Urbana-Champaign taught by Lena Fowalczny in Fall 2016. Since its upload, it has received 10 views. For similar materials see Elementary Linear Algebra in Mathematics at University of Illinois at Urbana-Champaign.

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Date Created: 09/07/16

Matrices REF + RREF Wednesday, September 7, 2016 3:07 PM Review Problem using Gaussian Elimination and Backward Substitution -2 6 -4 -6 -4 15 -2 0 5 -9 24 45 Remember for Gaussian Elimination, the goal is to put the matrix in REF To do this, make the first coefficient of the first row = 1 by multiplying by - 1/2 New Matrix 1 -3 2 3 -4 15 -2 0 5 -9 24 45 Next make all the numbers underneath the 1 we just made into zeros. This is done by multiplying the first row and adding it to another row 4R1 + R2 4 -12 8 12 New Matrix 1 -3 2 3 0 3 6 12 5 -9 24 45 Next do the same thing with the third row -5R1 + R3 -5 15 -10 -15 New Matrix 1 -3 2 3 0 3 6 12 0 6 14 30 Now we want to make the 6 in R3 also a zero and we do this by multiplying row 2 by -2 and adding it to R3 -2R2 + R3 0 -6 -12 -24 New Matrix 1 -3 2 3 0 3 6 12 0 0 2 6 Now we need 1's in R2 and R3 and we do this by dividing R2 by 3 and R3 by 2 REF Matrix 1 -3 2 3 0 1 2 4 0 0 1 3 Now use backwards substitution to get the answer From Row 3 Z = 3 From Row 2 Y + 2z = 4 Y + 6 = 4 Y = -2 From Row 1 X - 3y + 2z = 3 X + 6 + 6 = 3 X=-9 X = -9 Y = -2 Z = 3 Introducing Reduced Row Echelon Form (RREF) Gauss Jordan Elimination is what puts matrices in RREF. This an example of a matrix in Reduced Row Echelon Form 1 0 0 3 0 1 0 4 0 0 1 -2 The way to put a matrix into RREF begins with putting the matrix in REF This is the REF matrix we had earlier 1 -3 2 3 0 1 2 4 0 0 1 3 Next, the goal will be to make everything a zero above the 1 in Row 3 Column 3 (highlighted above in red). -2R3 + R2 0 0 -2 -6 New Matrix 1 -3 2 3 0 1 0 -2 0 0 1 3 -2R3 + R1 0 0 -2 -6 New Matrix 1 -3 0 -3 0 1 0 -2 0 0 1 3 The next goal is to get rid of the -3 in Row 1 Column 2. 3R2 + R1 0 3 0 -6 New Matrix 1 0 0 -9 0 1 0 -2 0 0 1 3 As you can see there is matrix reads off the same answer as before X = -9 Y = -2 Z = 3 The purpose of putting a matrix into RREF is to bypass using backwards substitution. So instead, of doing any substitution, this method uses only elimination.

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