Find Reduced Row Echelon Form [[1,2,3],[x,y,z]]

Problem

[[1,2,3],[x,y,z]]

Solution

1. Identify the first pivot in the first row, which is the 1 in the first column.

2. Eliminate the entry below the first pivot by performing the row operation (R_2)⇒(R_2)−x*(R_1)

[[1,2,3],[x−x(1),y−x(2),z−x(3)]]

[[1,2,3],[0,y−2*x,z−3*x]]

3. Normalize the second row to create the second pivot by dividing (R_2) by y−2*x assuming y−2*x≠0

[[1,2,3],[0,1,(z−3*x)/(y−2*x)]]

4. Eliminate the entry above the second pivot by performing the row operation (R_1)⇒(R_1)−2*(R_2)

[[1,2−2*(1),3−2*((z−3*x)/(y−2*x))],[0,1,(z−3*x)/(y−2*x)]]

5. Simplify the expression in the first row, third column.

3−(2*z−6*x)/(y−2*x)=(3*(y−2*x)−(2*z−6*x))/(y−2*x)

(3*y−6*x−2*z+6*x)/(y−2*x)=(3*y−2*z)/(y−2*x)

Final Answer

rref*[[1,2,3],[x,y,z]]=[[1,0,(3*y−2*z)/(y−2*x)],[0,1,(z−3*x)/(y−2*x)]]

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