Simplify the Matrix

Problem

[[1,0,9,0,−7],[0,1,7,0,7],[0,0,0,1,4],[0,0,0,0,−3]]

Solution

1. Identify the goal to simplify the matrix into its reduced row echelon form (RREF).

2. Normalize the fourth row by dividing by −3 to create a leading 1 in the last row.

[[1,0,9,0,−7],[0,1,7,0,7],[0,0,0,1,4],[0,0,0,0,1]]

3. Eliminate the entry in the third row, fifth column by performing (R_3)→(R_3)−4*(R_4)

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

4. Eliminate the entry in the second row, fifth column by performing (R_2)→(R_2)−7*(R_4)

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

5. Eliminate the entry in the first row, fifth column by performing (R_1)→(R_1)+7*(R_4)

[[1,0,9,0,0],[0,1,7,0,0],[0,0,0,1,0],[0,0,0,0,1]]

Final Answer

[[1,0,9,0,−7],[0,1,7,0,7],[0,0,0,1,4],[0,0,0,0,−3]]=[[1,0,9,0,0],[0,1,7,0,0],[0,0,0,1,0],[0,0,0,0,1]]

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