Simplify the Matrix [[1,0,0],[0,r-2,2],[0,s-5,r+2],[0,0,3]]

Problem

[[1,0,0],[0,r−2,2],[0,s−5,r+2],[0,0,3]]

Solution

1. Identify the goal of simplifying a matrix, which typically involves performing elementary row operations to reach Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).

2. Observe that the first column is already in the desired form with a leading 1 in the first row and zeros below it.

3. Swap the fourth row with the second row to move the constant entry to a higher position, which can simplify the elimination process.

[[1,0,0],[0,0,3],[0,s−5,r+2],[0,r−2,2]]

4. Normalize the new second row by multiplying by 1/3 to create a leading 1 in the third column.

[[1,0,0],[0,0,1],[0,s−5,r+2],[0,r−2,2]]

5. Eliminate the entries in the third column for the third and fourth rows using the new second row.

(R_3)→(R_3)−(r+2)*(R_2)

(R_4)→(R_4)−2*(R_2)

[[1,0,0],[0,0,1],[0,s−5,0],[0,r−2,0]]

6. Swap the second and third rows to organize the leading entries.

[[1,0,0],[0,s−5,0],[0,0,1],[0,r−2,0]]

7. Eliminate the entry in the fourth row using the second row, assuming s≠5

(R_4)→(R_4)−(r−2)/(s−5)*(R_2)

[[1,0,0],[0,s−5,0],[0,0,1],[0,0,0]]

8. Scale the second row by 1/(s−5) to achieve the final Reduced Row Echelon Form.

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

Final Answer

[[1,0,0],[0,r−2,2],[0,s−5,r+2],[0,0,3]]→[[1,0,0],[0,1,0],[0,0,1],[0,0,0]]

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