Find Reduced Row Echelon Form

Problem

[[−2/6.5,2/7,0,0],[−6/6.5,−6/7,6/6.5,0],[1.5/6.5,3/7,2.5/6.5,−300]]

Solution

1. Simplify the fractions and decimals in the matrix to make calculations easier.

A=[[−4/13,2/7,0,0],[−12/13,−6/7,12/13,0],[3/13,3/7,5/13,−300]]

2. Normalize the first row by multiplying (R_1) by −13/4 to create a leading 1.

(R_1)→−13/4*(R_1)=[[1,−13/14,0,0]]

3. Eliminate the first column entries below the leading 1 by performing (R_2)→(R_2)+12/13*(R_1) and (R_3)→(R_3)−3/13*(R_1)

(R_2)→[[0,−12/7,12/13,0]]

(R_3)→[[0,9/14,5/13,−300]]

4. Normalize the second row by multiplying (R_2) by −7/12 to create the next leading 1.

(R_2)→−7/12*(R_2)=[[0,1,−7/13,0]]

5. Eliminate the second column entries above and below the leading 1 by performing (R_1)→(R_1)+13/14*(R_2) and (R_3)→(R_3)−9/14*(R_2)

(R_1)→[[1,0,−1/2,0]]

(R_3)→[[0,0,10/13,−300]]

6. Normalize the third row by multiplying (R_3) by 13/10 to create the final leading 1.

(R_3)→13/10*(R_3)=[[0,0,1,−390]]

7. Eliminate the third column entries above the leading 1 by performing (R_1)→(R_1)+1/2*(R_3) and (R_2)→(R_2)+7/13*(R_3)

(R_1)→[[1,0,0,−195]]

(R_2)→[[0,1,0,−210]]

Final Answer

rref*[[−2/6.5,2/7,0,0],[−6/6.5,−6/7,6/6.5,0],[1.5/6.5,3/7,2.5/6.5,−300]]=[[1,0,0,−195],[0,1,0,−210],[0,0,1,−390]]

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