Find Reduced Row Echelon Form [[3,-2,7,63],[2,1,-21,28],[1,4,-49,-7]]

Problem

[[3,−2,7,63],[2,1,−21,28],[1,4,−49,−7]]

Solution

1. Swap rows to place a 1 in the first pivot position by moving the third row to the first row.

(R_1)↔(R_3)

[[1,4,−49,−7],[2,1,−21,28],[3,−2,7,63]]

2. Eliminate the entries below the first pivot by subtracting multiples of the first row from the second and third rows.

(R_2)−2*(R_1)→(R_2)

(R_3)−3*(R_1)→(R_3)

[[1,4,−49,−7],[0,−7,77,42],[0,−14,154,84]]

3. Normalize the second row by dividing it by −7 to create a leading 1

−1/7*(R_2)→(R_2)

[[1,4,−49,−7],[0,1,−11,−6],[0,−14,154,84]]

4. Eliminate the entries above and below the second pivot using the second row.

(R_1)−4*(R_2)→(R_1)

(R_3)+14*(R_2)→(R_3)

[[1,0,−5,17],[0,1,−11,−6],[0,0,0,0]]

Final Answer

rref*[[3,−2,7,63],[2,1,−21,28],[1,4,−49,−7]]=[[1,0,−5,17],[0,1,−11,−6],[0,0,0,0]]

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