Math

QuestionRow reduce the matrix below to reduced echelon form and identify pivot positions.
[123456786789] \left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]

Studdy Solution

STEP 1

Assumptions1. The given matrix is a3x4 matrix. . Row reduction is performed using elementary row operations.
3. The pivot positions are the leading1's in each row of the reduced echelon form.
4. The pivot columns are the columns in the original matrix that correspond to the pivot positions in the reduced echelon form.

STEP 2

Begin by subtracting5 times the first row from the second row, and6 times the first row from the third row. This will create zeros in the first column of the second and third rows.
[12404812051015]\left[\begin{array}{rrrr} 1 &2 & &4 \\ 0 & -4 & -8 & -12 \\ 0 & -5 & -10 & -15\end{array}\right]

STEP 3

Next, multiply the second row by -1/ and the third row by -1/5 to get leading1's in the second and third rows.
[12301230123]\left[\begin{array}{rrrr} 1 &2 &3 & \\ 0 &1 &2 &3 \\ 0 &1 &2 &3\end{array}\right]

STEP 4

Subtract the second row from the third row to create a zero in the second column of the third row.
[123401230000]\left[\begin{array}{rrrr} 1 &2 &3 &4 \\ 0 &1 &2 &3 \\ 0 &0 &0 &0\end{array}\right]

STEP 5

Finally, subtract2 times the second row from the first row to create a zero in the second column of the first row. This is the reduced echelon form of the matrix.
[101201230000]\left[\begin{array}{rrrr} 1 &0 & -1 & -2 \\ 0 &1 &2 &3 \\ 0 &0 &0 &0\end{array}\right]

STEP 6

The pivot positions in the final matrix are the first positions in the first and second rows, so the answer is A.

STEP 7

The pivot positions in the original matrix are also the first positions in the first and second rows, so the answer is A.

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