Simplify the Matrix

Problem

[[1,0,0],[0,1,3],[0,0,1]]*[[1,0,−8],[0,1,0],[0,0,1]]*[[1,0,0],[0,1,0],[0,−1,1]]*[[1,−2,0],[0,1,0],[0,0,1]]*[[1,0,0],[0,1/2,0],[0,0,1]]*[[1,0,0],[0,1,0],[2,0,1]]*[[1,0,0],[−3,1,0],[0,0,1]]

Solution

1. Identify the matrices as elementary matrices representing row operations. Let the product be A=(E_7)*(E_6)*(E_5)*(E_4)*(E_3)*(E_2)*(E_1)

2. Multiply the first two matrices from the right, (E_2)*(E_1) which corresponds to (R_2)→(R_2)−3*(R_1) followed by (R_3)→(R_3)+2*(R_1)

(E_2)*(E_1)=[[1,0,0],[−3,1,0],[2,0,1]]

3. Multiply the result by (E_3) which corresponds to (R_2)→1/2*(R_2)

(E_3)*((E_2)*(E_1))=[[1,0,0],[−3/2,1/2,0],[2,0,1]]

4. Multiply the result by (E_4) which corresponds to (R_1)→(R_1)−2*(R_2)

(E_4)*((E_3)*(E_2)*(E_1))=[[4,−1,0],[−3/2,1/2,0],[2,0,1]]

5. Multiply the result by (E_5) which corresponds to (R_2)→(R_2)−(R_3)

(E_5)*((E_4)*(E_3)*(E_2)*(E_1))=[[4,−1,0],[−7/2,1/2,−1],[2,0,1]]

6. Multiply the result by (E_6) which corresponds to (R_1)→(R_1)−8*(R_3)

(E_6)*((E_5)*(E_4)*(E_3)*(E_2)*(E_1))=[[−12,−1,−8],[−7/2,1/2,−1],[2,0,1]]

7. Multiply the result by (E_7) which corresponds to (R_2)→(R_2)+3*(R_3)

(E_7)*((E_6)*(E_5)*(E_4)*(E_3)*(E_2)*(E_1))=[[−12,−1,−8],[5/2,1/2,2],[2,0,1]]

Final Answer

[[1,0,0],[0,1,3],[0,0,1]]*[[1,0,−8],[0,1,0],[0,0,1]]*[[1,0,0],[0,1,0],[0,−1,1]]*[[1,−2,0],[0,1,0],[0,0,1]]*[[1,0,0],[0,1/2,0],[0,0,1]]*[[1,0,0],[0,1,0],[2,0,1]]*[[1,0,0],[−3,1,0],[0,0,1]]=[[−12,−1,−8],[5/2,1/2,2],[2,0,1]]

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