Find Reduced Row Echelon Form

Problem

[[3*a,−2*b,c,12],[3*a,−2*b,c,−2],[3*a,−2*b,c,11]]

Solution

1. Subtract the first row from the second row ((R_2)−(R_1)→(R_2) to eliminate the variables in the second row.

[[3*a,−2*b,c,12],[0,0,0,−14],[3*a,−2*b,c,11]]

2. Subtract the first row from the third row ((R_3)−(R_1)→(R_3) to eliminate the variables in the third row.

[[3*a,−2*b,c,12],[0,0,0,−14],[0,0,0,−1]]

3. Normalize the second row by dividing by −14 (−1/14*(R_2)→(R_2) to create a leading one.

[[3*a,−2*b,c,12],[0,0,0,1],[0,0,0,−1]]

4. Eliminate the constant in the third row by adding the second row to the third row ((R_3)+(R_2)→(R_3).

[[3*a,−2*b,c,12],[0,0,0,1],[0,0,0,0]]

5. Eliminate the constant in the first row by subtracting 12 times the second row from the first row ((R_1)−12*(R_2)→(R_1).

[[3*a,−2*b,c,0],[0,0,0,1],[0,0,0,0]]

6. Divide the first row by 3*a (1/(3*a)*(R_1)→(R_1), assuming a≠0 to set the first leading coefficient to 1.

[[1,−(2*b)/(3*a),c/(3*a),0],[0,0,0,1],[0,0,0,0]]

Final Answer

rref*[[3*a,−2*b,c,12],[3*a,−2*b,c,−2],[3*a,−2*b,c,11]]=[[1,−(2*b)/(3*a),c/(3*a),0],[0,0,0,1],[0,0,0,0]]

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