Find Reduced Row Echelon Form [[a,b,c,1000],[100a,200b,500c,300000]]

Problem

[[a,b,c,1000],[100*a,200*b,500*c,300000]]

Solution

1. Divide the second row by 100 to simplify the coefficients.

(R_2)→1/100*(R_2)

[[a,b,c,1000],[a,2*b,5*c,3000]]

2. Subtract the first row from the second row to eliminate the a term in the second row.

(R_2)→(R_2)−(R_1)

[[a,b,c,1000],[0,b,4*c,2000]]

3. Subtract the new second row from the first row to eliminate the b term in the first row.

(R_1)→(R_1)−(R_2)

[[a,0,−3*c,−1000],[0,b,4*c,2000]]

4. Divide the first row by a and the second row by b to obtain leading ones, assuming a≠0 and b≠0

(R_1)→1/a*(R_1)

(R_2)→1/b*(R_2)

[[1,0,−(3*c)/a,−1000/a],[0,1,(4*c)/b,2000/b]]

Final Answer

rref*([a,b,c,1000],[100*a,200*b,500*c,300000])=[[1,0,−(3*c)/a,−1000/a],[0,1,(4*c)/b,2000/b]]

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