Find Pivot Positions and Pivot Columns B=[[1,3,6],[-6,10,12],[3,-3,-2]]

Problem

B=[[1,3,6],[−6,10,12],[3,−3,−2]]

Solution

1. Identify the first pivot in the first row, first column, which is 1 Use row operations to create zeros below it.

2. Perform the row operations (R_2)+6*(R_1)→(R_2) and (R_3)−3*(R_1)→(R_3)

B∼[[1,3,6],[0,28,48],[0,−12,−20]]

3. Simplify the second row by dividing by 4 to make calculations easier ((R_2)÷4→(R_2).

B∼[[1,3,6],[0,7,12],[0,−12,−20]]

4. Identify the second pivot in the second row, second column, which is 7 Use row operations to create a zero below it.

5. Perform the row operation (R_3)+12/7*(R_2)→(R_3)

B∼[[1,3,6],[0,7,12],[0,0,4/7]]

6. Identify the pivot positions. The matrix is now in echelon form. The leading entries (pivots) are in row 1 column 1, row 2 column 2, and row 3 column 3.

7. Determine the pivot columns based on the positions of the leading entries in the echelon form.

Final Answer

Pivot Positions: (1, 1), (2, 2), (3, 3)

Pivot Columns: Column 1, Column 2, Column 3

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