Example

Example: Performing Matrix Row Operations

Consider the augmented matrix: [6βˆ’52321βˆ’453βˆ’31βˆ’1]\left[ \begin{array}{ccc|c} 6 & -5 & 2 & 3 \\\\ 2 & 1 & -4 & 5 \\\\ 3 & -3 & 1 & -1 \end{array} \right] We can perform various row operations independently to manipulate this matrix:

  • Interchanging rows: Swapping row 22 and row 33 yields: [6βˆ’5233βˆ’31βˆ’121βˆ’45]\left[ \begin{array}{ccc|c} 6 & -5 & 2 & 3 \\\\ 3 & -3 & 1 & -1 \\\\ 2 & 1 & -4 & 5 \end{array} \right]
  • Multiplying a row by a constant: Multiplying row 22 by 55 (denoted 5R25R_2) yields: [6βˆ’523105βˆ’20253βˆ’31βˆ’1]\left[ \begin{array}{ccc|c} 6 & -5 & 2 & 3 \\\\ 10 & 5 & -20 & 25 \\\\ 3 & -3 & 1 & -1 \end{array} \right]
  • Adding a multiple of one row to another: Multiplying row 33 by βˆ’2-2 and adding it to row 11 (denoted βˆ’2R3+R1-2R_3 + R_1) yields: [010521βˆ’453βˆ’31βˆ’1]\left[ \begin{array}{ccc|c} 0 & 1 & 0 & 5 \\\\ 2 & 1 & -4 & 5 \\\\ 3 & -3 & 1 & -1 \end{array} \right]

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Updated 2026-05-26

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