Find Reduced Row Echelon Form [[1,1,1],[1,2,k],[1,4,k^2]]

Problem

[[1,1,1],[1,2,k],[1,4,k2]]

Solution

1. Eliminate the first entries in the second and third rows by subtracting the first row from both.

(R_2)←(R_2)−(R_1)

(R_3)←(R_3)−(R_1)

[[1,1,1],[0,1,k−1],[0,3,k2−1]]

2. Eliminate the second entry in the third row by subtracting 3 times the second row from the third row.

(R_3)←(R_3)−3*(R_2)

[[1,1,1],[0,1,k−1],[0,0,k2−3*k+2]]

3. Factor the expression in the third row to identify potential pivots.

k2−3*k+2=(k−1)*(k−2)

[[1,1,1],[0,1,k−1],[0,0,(k−1)*(k−2)]]

4. Normalize the third row by dividing by (k−1)*(k−2) assuming k≠1 and k≠2

(R_3)←1/((k−1)*(k−2))*(R_3)

[[1,1,1],[0,1,k−1],[0,0,1]]

5. Eliminate the entries above the pivot in the third column.

(R_2)←(R_2)−(k−1)*(R_3)

(R_1)←(R_1)−(R_3)

[[1,1,0],[0,1,0],[0,0,1]]

6. Eliminate the entry above the pivot in the second column.

(R_1)←(R_1)−(R_2)

[[1,0,0],[0,1,0],[0,0,1]]

Final Answer

rref*[[1,1,1],[1,2,k],[1,4,k2]]=[[1,0,0],[0,1,0],[0,0,1]],k∉{1,2}

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