Simplify the Matrix [[1,3,0,4],[0,2,-1,5],[3,0,1,-2]]

Problem

[[1,3,0,4],[0,2,−1,5],[3,0,1,−2]]

Solution

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

2. Eliminate the first entry in the third row by performing the row operation (R_3)←(R_3)−3*(R_1)

[[1,3,0,4],[0,2,−1,5],[3−3*(1),0−3*(3),1−3*(0),−2−3*(4)]]

[[1,3,0,4],[0,2,−1,5],[0,−9,1,−14]]

3. Normalize the second row by dividing by 2 to create a leading 1.

(R_2)←1/2*(R_2)

[[1,3,0,4],[0,1,−0.5,2.5],[0,−9,1,−14]]

4. Eliminate the entries above and below the leading 1 in the second column using (R_1)←(R_1)−3*(R_2) and (R_3)←(R_3)+9*(R_2)

[[1,0,1.5,−3.5],[0,1,−0.5,2.5],[0,0,−3.5,8.5]]

5. Normalize the third row by dividing by −3.5 to create a leading 1.

(R_3)←−2/7*(R_3)

[[1,0,1.5,−3.5],[0,1,−0.5,2.5],[0,0,1,−17/7]]

6. Eliminate the entries above the leading 1 in the third column using (R_1)←(R_1)−1.5*(R_3) and (R_2)←(R_2)+0.5*(R_3)

(R_1)←(R_1)−3/2*(R_3)⇒−3.5−3/2*(−17/7)=−7/2+51/14=1/7

(R_2)←(R_2)+1/2*(R_3)⇒2.5+1/2*(−17/7)=5/2−17/14=9/7

[[1,0,0,1/7],[0,1,0,9/7],[0,0,1,−17/7]]

Final Answer

[[1,3,0,4],[0,2,−1,5],[3,0,1,−2]]=[[1,0,0,1/7],[0,1,0,9/7],[0,0,1,−17/7]]

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