Simplify the Matrix a=[[1,-2,4],[2,-5,12],[0,2,-10]]

Problem

A=[[1,−2,4],[2,−5,12],[0,2,−10]]

Solution

1. Identify the goal to simplify the matrix by converting it to Row Echelon Form (REF) using Gaussian elimination.

2. Eliminate the entry in the second row, first column by performing the row operation (R_2)←(R_2)−2*(R_1)

[[1,−2,4],[2−2*(1),−5−2*(−2),12−2*(4)],[0,2,−10]]=[[1,−2,4],[0,−1,4],[0,2,−10]]

3. Eliminate the entry in the third row, second column by performing the row operation (R_3)←(R_3)+2*(R_2)

[[1,−2,4],[0,−1,4],[0+2*(0),2+2*(−1),−10+2*(4)]]=[[1,−2,4],[0,−1,4],[0,0,−2]]

4. Normalize the rows by multiplying (R_2) by −1 and (R_3) by −1/2 to obtain leading ones.

[[1,−2,4],[0,1,−4],[0,0,1]]

5. Reduce to Reduced Row Echelon Form (RREF) by eliminating the entries above the pivots. First, perform (R_2)←(R_2)+4*(R_3) and (R_1)←(R_1)−4*(R_3)

[[1,−2,0],[0,1,0],[0,0,1]]

6. Finalize the RREF by performing (R_1)←(R_1)+2*(R_2)

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

Final Answer

[[1,−2,4],[2,−5,12],[0,2,−10]]⇒[[1,0,0],[0,1,0],[0,0,1]]

Want more problems? Check here!