Skip to content

Matrix Inverse Explorer

Run the Matrix Inverse Explorer MicroSim Fullscreen

Edit the Matrix Inverse Explorer MicroSim with the p5.js editor

You can include this MicroSim on your website using the following iframe:

1
<iframe src="https://dmccreary.github.io/linear-algebra/sims/matrix-inverse/main.html" height="452px" scrolling="no"></iframe>

Description

The matrix inverse generalizes division to matrices. For a square matrix A, its inverse A⁻¹ (if it exists) satisfies AA⁻¹ = A⁻¹A = I, where I is the identity matrix. This MicroSim lets you explore matrix inversion interactively.

Key Features:

  • Real-Time Computation: See the inverse update instantly for random matrices
  • Verification Display: Watch AA⁻¹ = I computed live
  • Determinant Indicator: Color-coded display shows invertibility status
  • Singularity Exploration: Make matrices singular or approach singularity smoothly
  • Formula Display: See the 2×2 inverse formula applied

The 2×2 Inverse Formula

For a 2×2 matrix:

\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

The inverse is:

\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]

Key insight: The inverse exists if and only if det(A) = ad - bc ≠ 0.

Singular Matrices

A matrix is singular (not invertible) when:

  • det(A) = 0
  • The columns are linearly dependent
  • The matrix maps some non-zero vector to zero

The MicroSim shows this by turning red when you click "Make Singular" or slide toward singularity.

Lesson Plan

Learning Objectives

After using this MicroSim, students will be able to:

  1. Apply the 2×2 inverse formula to compute A⁻¹
  2. Verify that AA⁻¹ = I for invertible matrices
  3. Identify singular matrices by their zero determinant
  4. Explain why singular matrices have no inverse

Guided Exploration (5-7 minutes)

  1. Start with Default: Observe the invertible matrix and verify AA⁻¹ = I
  2. Click Randomize: Generate new matrices and check determinants
  3. Make Singular: Click the button and observe the warning
  4. Approach Singularity: Use the slider to see determinant approach zero

Discussion Points

  • What happens to the inverse entries as det(A) approaches zero?
  • Why does "Make Singular" make row 2 proportional to row 1?
  • How does this connect to solving systems of equations?

Assessment Questions

  1. For A = [[3, 1], [2, 1]], compute det(A) and A⁻¹.
  2. Why can't you divide by a matrix the way you divide by a number?
  3. If det(A) = 0.001, is A technically invertible? Is it practically invertible?

References