Find Reduced Row Echelon Form

Problem

[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]

Solution

1. Eliminate the first entries of the second, third, and fourth rows by performing row operations using the first row.

(R_2)→(R_2)−5*(R_1)

(R_3)→(R_3)−9*(R_1)

(R_4)→(R_4)−13*(R_1)

[[1,2,3,4],[0,−4,−8,−12],[0,−8,−16,−24],[0,−12,−24,−36]]

2. Normalize the second row by dividing it by −4 to create a leading 1.

(R_2)→−1/4*(R_2)

[[1,2,3,4],[0,1,2,3],[0,−8,−16,−24],[0,−12,−24,−36]]

3. Eliminate the entries in the second column for all other rows using the second row.

(R_1)→(R_1)−2*(R_2)

(R_3)→(R_3)+8*(R_2)

(R_4)→(R_4)+12*(R_2)

[[1,0,−1,−2],[0,1,2,3],[0,0,0,0],[0,0,0,0]]

4. Identify that the matrix is now in Reduced Row Echelon Form (RREF) as all leading ones are the only non-zero entries in their columns and all zero rows are at the bottom.

Final Answer

rref*[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]=[[1,0,−1,−2],[0,1,2,3],[0,0,0,0],[0,0,0,0]]

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