Find the Rank [[6,8,9],[3,5,7],[9,4,2]]

Problem

rank(6)

Solution

1. Set up the matrix A

A=[[6,8,9],[3,5,7],[9,4,2]]

2. Swap the first and second rows ((R_1)↔(R_2) to get a smaller leading entry in the first row.

[[3,5,7],[6,8,9],[9,4,2]]

3. Eliminate the first entries of the second and third rows by performing (R_2)−2*(R_1)→(R_2) and (R_3)−3*(R_1)→(R_3)

[[3,5,7],[0,−2,−5],[0,−11,−19]]

4. Eliminate the second entry of the third row by performing (R_3)−11/2*(R_2)→(R_3)

[[3,5,7],[0,−2,−5],[0,0,17/2]]

5. Identify the number of non-zero rows in the row-echelon form. Since there are 3 non-zero rows, the rank is 3.

Final Answer

rank(6)=3

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