Find Reduced Row Echelon Form

Problem

[[1+√(,2),0,1],[3,√(,2),−3],[1,0,1+√(,2)]]

Solution

1. Identify the initial matrix A

A=[[1+√(,2),0,1],[3,√(,2),−3],[1,0,1+√(,2)]]

2. Swap the first and third rows to make the leading element of the first row simpler.

(R_1)↔(R_3)⇒[[1,0,1+√(,2)],[3,√(,2),−3],[1+√(,2),0,1]]

3. Eliminate the first element of the second row by subtracting 3 times the first row from the second row.

(R_2)−3*(R_1)→(R_2)⇒[[1,0,1+√(,2)],[0,√(,2),−6−3√(,2)],[1+√(,2),0,1]]

4. Eliminate the first element of the third row by subtracting (1+√(,2)) times the first row from the third row.

(R_3)−(1+√(,2))*(R_1)→(R_3)⇒[[1,0,1+√(,2)],[0,√(,2),−6−3√(,2)],[0,0,1−(1+√(,2))2]]

5. Simplify the element in the third row, third column.

1−(1+2√(,2)+2)=−2−2√(,2)

[[1,0,1+√(,2)],[0,√(,2),−6−3√(,2)],[0,0,−2−2√(,2)]]

6. Normalize the second and third rows by dividing each row by its leading non-zero coefficient.

1/√(,2)*(R_2)→(R_2)⇒[[1,0,1+√(,2)],[0,1,−3√(,2)−3],[0,0,−2−2√(,2)]]

1/(−2−2√(,2))*(R_3)→(R_3)⇒[[1,0,1+√(,2)],[0,1,−3√(,2)−3],[0,0,1]]

7. Eliminate the third column elements in the first and second rows using the third row.

(R_1)−(1+√(,2))*(R_3)→(R_1)⇒[[1,0,0],[0,1,−3√(,2)−3],[0,0,1]]

(R_2)−(−3√(,2)−3)*(R_3)→(R_2)⇒[[1,0,0],[0,1,0],[0,0,1]]

Final Answer

rref*[[1+√(,2),0,1],[3,√(,2),−3],[1,0,1+√(,2)]]=[[1,0,0],[0,1,0],[0,0,1]]

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