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Determinant Properties Explorer

Run the Determinant Properties Explorer Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This explorer helps you understand how different operations affect the determinant of a matrix.

Key Properties:

Operation Effect on Determinant
Swap rows \(\det(A') = -\det(A)\)
Scale row by k \(\det(A') = k \cdot \det(A)\)
Add multiple of one row to another \(\det(A') = \det(A)\) (unchanged)
Transpose \(\det(A^T) = \det(A)\) (unchanged)

How to Use

  1. Click operation buttons to apply different transformations
  2. Adjust k slider to change the scaling/addition factor
  3. Compare matrices - see original (left) and modified (right)
  4. Watch parallelogram area change in the geometric view
  5. Click Reset to restore the original matrix

Embedding

1
<iframe src="https://dmccreary.github.io/linear-algebra/sims/det-properties/main.html" height="502px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Predict how row operations change the determinant
  2. Explain why adding row multiples preserves the determinant
  3. Connect algebraic properties to geometric area changes

Suggested Activities

  1. Verify properties: Apply each operation and check the relationship holds
  2. Chain operations: What happens if you swap, then scale, then swap back?
  3. Find invariants: Which operations preserve |det|?

Assessment Questions

  1. If det(A) = 6, what is det(A) after scaling row 1 by 3?
  2. You perform 5 row swaps. Is det(A') positive or negative if det(A) > 0?
  3. Why doesn't adding row multiples change the determinant geometrically?

References