Simplify the Matrix

Problem

[[5,−10,25,−20],[1,−2,5,−4],[3,−6,15,12]]

Solution

1. Identify the goal to simplify the matrix by performing row reduction to reach Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).

2. Swap the first and second rows to obtain a leading 1 in the first row.

(R_1)↔(R_2)

[[1,−2,5,−4],[5,−10,25,−20],[3,−6,15,12]]

3. Eliminate the first entries of the second and third rows by using row operations based on the first row.

(R_2)−5*(R_1)→(R_2)

(R_3)−3*(R_1)→(R_3)

[[1,−2,5,−4],[0,0,0,0],[0,0,0,24]]

4. Swap the second and third rows to move the row of zeros to the bottom.

(R_2)↔(R_3)

[[1,−2,5,−4],[0,0,0,24],[0,0,0,0]]

5. Normalize the second row by dividing by 24 to create a leading 1

1/24*(R_2)→(R_2)

[[1,−2,5,−4],[0,0,0,1],[0,0,0,0]]

6. Eliminate the entry above the leading 1 in the second column to reach Reduced Row Echelon Form.

(R_1)+4*(R_2)→(R_1)

[[1,−2,5,0],[0,0,0,1],[0,0,0,0]]

Final Answer

[[5,−10,25,−20],[1,−2,5,−4],[3,−6,15,12]]=[[1,−2,5,0],[0,0,0,1],[0,0,0,0]]

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