Simplify the Matrix [[1,0,0,5],[3,1,0,25/3],[0,3,0,20]]

Problem

[[1,0,0,5],[3,1,0,25/3],[0,3,0,20]]

Solution

1. Identify the goal to simplify the matrix into Reduced Row Echelon Form (RREF) using Gaussian elimination.

2. Eliminate the first entry in the second row by performing the row operation (R_2)−3*(R_1)⇒(R_2)

(R_2)=[[3,1,0,25/3]]−3*[[1,0,0,5]]=[[0,1,0,−20/3]]

3. Update the matrix with the new second row.

[[1,0,0,5],[0,1,0,−20/3],[0,3,0,20]]

4. Eliminate the second entry in the third row by performing the row operation (R_3)−3*(R_2)⇒(R_3)

(R_3)=[[0,3,0,20]]−3*[[0,1,0,−20/3]]=[[0,0,0,40]]

5. Normalize the third row by dividing by 40 to create a leading one, (R_3)/40⇒(R_3)

[[1,0,0,5],[0,1,0,−20/3],[0,0,0,1]]

6. Eliminate the constants in the fourth column above the leading one in the third row using (R_1)−5*(R_3)⇒(R_1) and (R_2)+20/3*(R_3)⇒(R_2)

(R_1)=[[1,0,0,5]]−5*[[0,0,0,1]]=[[1,0,0,0]]

(R_2)=[[0,1,0,−20/3]]+20/3*[[0,0,0,1]]=[[0,1,0,0]]

Final Answer

[[1,0,0,5],[3,1,0,25/3],[0,3,0,20]]=[[1,0,0,0],[0,1,0,0],[0,0,0,1]]

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