Simplify the Matrix [[1,0,0],[0,r-6,6],[0,s-5,r+6],[0,0,7]]

Problem

[[1,0,0],[0,r−6,6],[0,s−5,r+6],[0,0,7]]

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 simplified with a leading 1 in the first row and zeros below it.

3. Eliminate the entry in the fourth row by using row operations. Since the fourth row is [[0,0,7]] we can divide the fourth row by 7 to create a leading 1

(R_4)→1/7*(R_4)

[[1,0,0],[0,r−6,6],[0,s−5,r+6],[0,0,1]]

4. Clear the entries above the leading 1 in the third column by using the new fourth row.

(R_2)→(R_2)−6*(R_4)

(R_3)→(R_3)−(r+6)*(R_4)

[[1,0,0],[0,r−6,0],[0,s−5,0],[0,0,1]]

5. Analyze the remaining 2×1 sub-block in the second column. To simplify further, we assume the system is consistent or we are seeking the simplest algebraic form. If r≠6 we can divide the second row by r−6

(R_2)→1/(r−6)*(R_2)

[[1,0,0],[0,1,0],[0,s−5,0],[0,0,1]]

6. Eliminate the entry in the third row using the second row.

(R_3)→(R_3)−(s−5)*(R_2)

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

7. Swap rows to reach standard Row Echelon Form.

(R_3)↔(R_4)

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

Final Answer

[[1,0,0],[0,r−6,6],[0,s−5,r+6],[0,0,7]]=[[1,0,0],[0,1,0],[0,0,1],[0,0,0]]

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