Simplify the Matrix

Problem

[[1,0.7,0.3,0],[−0.3,0.8,−0.5,0],[−0.15,−0.3,0.45,0]]

Solution

1. Identify the goal to simplify the matrix by performing row operations to reach Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).

2. Eliminate the entries below the first pivot in the first column. Add 0.3 times the first row to the second row, and add 0.15 times the first row to the third row.

(R_2)←(R_2)+0.3*(R_1)

(R_3)←(R_3)+0.15*(R_1)

[[1,0.7,0.3,0],[0,1.01,−0.41,0],[0,−0.195,0.495,0]]

3. Normalize the second row by dividing by the pivot element 1.01

(R_2)←(R_2)/1.01

[[1,0.7,0.3,0],[0,1,−41/101,0],[0,−0.195,0.495,0]]

4. Eliminate the entry below the second pivot. Add 0.195 times the second row to the third row.

(R_3)←(R_3)+0.195*(R_2)

[[1,0.7,0.3,0],[0,1,−0.40594,0],[0,0,0.41584,0]]

5. Normalize the third row by dividing by the pivot element 0.41584

(R_3)←(R_3)/0.41584

[[1,0.7,0.3,0],[0,1,−0.40594,0],[0,0,1,0]]

6. Eliminate the entries above the third pivot to reach RREF.

(R_2)←(R_2)+0.40594*(R_3)

(R_1)←(R_1)−0.3*(R_3)

[[1,0.7,0,0],[0,1,0,0],[0,0,1,0]]

7. Eliminate the entry above the second pivot.

(R_1)←(R_1)−0.7*(R_2)

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

Final Answer

[[1,0.7,0.3,0],[−0.3,0.8,−0.5,0],[−0.15,−0.3,0.45,0]]=[[1,0,0,0],[0,1,0,0],[0,0,1,0]]

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