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Row Operations Practice

Run the Row Operations MicroSim Fullscreen

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About This MicroSim

Practice applying the three elementary row operations on an augmented matrix. This tool helps build fluency with the operations used in Gaussian elimination.

The Three Row Operations:

Operation Notation Description
Row Swap R₁ ↔ R₂ Exchange two rows
Row Scaling k × Rᵢ → Rᵢ Multiply a row by a nonzero constant
Row Addition Rᵢ + k×Rⱼ → Rᵢ Add a multiple of one row to another

How to Use

  1. Select Operation Type: Choose Swap, Scale, or Add Multiple from the dropdown
  2. Choose Rows: Select which row(s) the operation affects
  3. Set Scalar: For scaling or adding, enter the multiplier value
  4. Apply: Click "Apply Operation" to execute
  5. Track Progress: View your operation history on the right
  6. Undo/Reset: Use Undo to step back or Reset to start over

Default System

The default matrix represents this system:

\[2x + y - z = 8$$ $$-3x - y + 2z = -11$$ $$-2x + y + 2z = -3\]

The solution is \(x = 2\), \(y = 3\), \(z = -1\).

Lesson Plan

Learning Objectives

After using this MicroSim, students will be able to:

  1. Execute the three elementary row operations correctly
  2. Understand how row operations preserve the solution set
  3. Develop intuition for which operations help reach row echelon form

Suggested Activities

  1. Reduce to REF: Use row operations to reduce the matrix to row echelon form
  2. Minimal Operations Challenge: Find the shortest sequence of operations to reach REF
  3. Reverse Engineering: Given an REF matrix, work backward to find an original system
  4. Create Zeros: Practice creating zeros below pivots using row addition

Tips for Efficient Reduction

  • Start with the leftmost column
  • Use row swaps to place a convenient pivot at the top
  • Scale to get a leading 1 (optional but often helpful)
  • Use row addition to create zeros below each pivot
  • Move right and repeat